Meta: these just happen to be the things they’ve written
These oaths produce paladins because these are the oaths that Wizards of the Coast has written up paladin abilities for. As soon as they—or a DM, for their particular campaign—write up paladin abilities for other oaths, those oaths will also suddenly also allow someone to become a paladin.
And, of course, in order to get any powers, you have to take the appropriate game-mechanical options, which come with some opportunity cost appropriate to the power you get. That is, you have to take paladin levels to get powers from an oath because that is you “paying” for those powers by not taking, say, barbarian at those same levels. Giving those powers away for free would be unfair.
Narrative: Mythopoeia, probably
The short answer is that you can’t be a paladin of some other oath because (enough of) the people of the Multiverse don’t believe in paladins of that thing (strongly enough). You need their belief to fuel your divine abilities. And you can’t tap into the power of that belief without the right spiritual preparation, which requires paladin training.
Evidence from D&D 5e
This interpretation suggests that the paladin’s oath doesn’t produce power—it allows the paladin to tap into an existing reserve of faith and belief, a sort of divine power not attached to any single deity. The descriptions of the oath in the Player’s Handbook seem to back this up: they emphasize the oath as a thing between the paladin and the divine power they wield:
|“a paladin’s oath is a powerful bond,”
||82, paladin intro
||Implies a connection to something else.
|“the oaths that grant them power,”
||82, paladin intro
||Only says the power comes through the oath, not where the power comes from.
|“[Paladins’] access to the Weave is mediated by divine power—[…]the sacred weight of a paladin’s oath,”
||206, sidebar: “The Weave of Magic”
||Positions the oath as “mediator,” not the ultimate source of power.
The oath is a connection to people’s belief in that which was sworn—there is power in making that connection, but most of the paladin’s abilities come from the belief.
Evidence from the wider D&D canon
This is also consistent with how divine power generally works in D&D—the entire theme revolves around tapping into something greater than yourself. (Manifesting power purely from one’s own personal determination and will is usually more in line with what D&D calls psionics.) It’s also consistent with how belief generally works in D&D—while the subject has never been covered specifically for 5e-style paladin oaths, plenty of other things work in a similar manner, right up to the gods themselves.
D&D is a partially mythopoetic setting, meaning that belief has real, tangible effect on reality, and certain things are what they are believed to be, because that’s what they’re believed to be. The Outer Planes are literally made from solid belief, and wouldn’t exist without that belief—that’s why they’re aligned, because the alignments are major groupings of belief that have coalesced into various planes.¹ Celestials, fiends, and other denizens of the Outer Planes—who have been known in earlier editions of D&D as “outsiders,” referring to their Outer Plane origins—are belief incarnate, as in fiends as “evil incarnate.” D&D’s gods take this to a logical extreme—they need people to believe not only in their portfolio, but also to believe in them, personally.
Nothing in 5e discusses this to any significant extent, to my knowledge. The Player’s Handbook mentions the Outer Planes as being “of thought and spirit,” which roughly means “belief,” but it isn’t explicit. The complex ways in which belief shapes the Outer Planes, such as the Knights Harmonium making an entire layer of Arcadia slide into Mechanus by being so lawful (and so distinctly not good), or parts of the Gate Towns around the Outlands shifting off into their respective planes and back as attitudes in the town get closer or further from that plane, goes unmentioned. These things are—presumably—still a part of the setting, but 5e hasn’t mentioned them (yet?).
This matters because 5e has changed the way paladins work; while paladins always involved oaths, in most editions of D&D, there was just one paladin oath, which all paladins took—and all paladins upheld as paragons of lawful goodness, specifically. Their power seemed to come from “the cosmic force of Good,” rather than any individual being. The Forgotten Realms had different oaths for different orders of paladins, but that’s because it—unlike most settings—required all paladins to be members of a knightly order sponsored by a specific patron deity, and then the paladins’ power came from that deity (so Mystra sponsored the Knights of the Mystic Fire, Tyr sponsored the Knights of Samular, etc. and to be a paladin you had to join one of these orders). But since 5e hasn’t talked about the effects of belief on reality in D&D, and has changed how paladins function, this answer necessarily requires a bit of synthesis.
Oath-breakers (and the Oath of Vengeance)
Oath-breakers deserve some special mention here, but mostly to just say they’re not actually all that special. Holding to one’s vows was, in antiquity, a primary virtue, and one that the paladin is largely based upon. This is part of why oaths have such power. It also means that there are strong beliefs about those who would break an oath. Breaking an oath isn’t just a betrayal of that specific thing, it’s a betrayal of all that is right and good. Plus, oath-breakers can likely still tap into the divine power that they once swore by, corrupting and tainting it.
(OP has also brought up the Oath of Vengeance here, but I think this is a more straightforward case: even if people at large don’t necessarily care—or even know—about whatever you’re seeking revenge for, they certainly have some very strong beliefs about the concept of vengeance, itself, and those who are driven by it. Those are the beliefs that such paladins tap into.)
Changing beliefs change the reality of D&D
This approach does mean that, in theory, if you got enough people to believe in your oath as something that one could be a paladin of, then you would be able to tap into that divine power and be a paladin of that thing. The rules, obviously, don’t discuss such a thing, since at that point you’re necessarily talking about a very custom thing that the DM is going to have to handle.
Narrative, alternate or addition: Overdeities, maybe
There are also some hints that paladins may—as in, it is possible—act differently in different campaign settings. This could be explained mythopoetically (local belief counts somewhat more than extraplanar belief, perhaps), but there is another force within D&D canon that might fit the bill better.
Overdeities: an overview
The Material Plane is divided into separate regions—we know them as campaign settings.² Each region has—presumably—an overdeity. We know that the Forgotten Realms’ overdeity is Ao, for example. We know much less about the overdeity for Dragonlance, but the High God is known to exist from their singular intervention in Krynn during the All-Saints War.
It is generally assumed among D&D lorists that each other campaign setting has their own overdeity—it allows for a simple, consistent explanation for why things are different from one campaign setting to another. Since even the few overdeities we know are extremely aloof from the settings they are ultimately in charge of—they take no worship, they espouse no beliefs, they intervene rarely if ever—it’s just assumed that the other overdeities are just a little more so and so we have no historical incidences of their interaction with the setting.
Thus, for example, we might assume that Eberron’s overdeity has some extremely strict rules about divine intervention—which leads to the difficulties being sure that deities even exist in that setting. This isn’t so different from Ao’s known rules about divine avatars, for example.
(To shift back to a meta view here for a moment, part of the reason why overdeities are so little-seen in D&D canon is because, on a meta level, they’re supposed to represent the DM, and D&D tries not to tell the DM what to do too much.)
Overdeities and paladins
Anyway, one of the things that could easily be within an overdeity’s purview is how paladins work in their setting, and what oaths can be a source of paladin power. We don’t have any examples of them doing so, but we do know that Ao could mess with how spells work in FR, requiring everyone to go through Mystra and the Weave (remember the Weave was an FR-only thing until 5e). Overdeities definitely mess with how belief works for the gods—cf. FR’s Wall of the Faithless as that pantheon’s answer to Ao’s rules on that subject. So they can probably tweak things for paladin oaths as well.
The reason I prefer the mythopoetic explanation, though, is that we have paladins outside of the Material Plane—and all evidence suggests that overdeities only exist for Material campaign settings, and that outside the Material Plane, no single god is above all the others. That makes it impossible for an overdeity to control planar paladins’ oaths. That said, it’s definitely plausible for an overdeity to modify—or even replace—the multiversal, mythopoetic rules within their own setting. We don’t really know the full details of what an overdeity can or can’t change (though again, to the extent they represent the DM, it might be anything), but most of what we know they have influenced has revolved around faith and the divine, so it’s reasonable to conclude that paladins are likely in their wheelhouse. So if an overdeity wants to let just anybody swear any oath and get divine power, that could very well be a thing they could do—but this requires a DM who makes that choice and is willing to support it by making the rules for it.
A question was raised about how these things affect planar travel, and/or are affected by planar travel. The answer is, by and large, that there is no interaction. Belief transcends planes: after all, the Outer Planes are literally built from—made of—the beliefs of those on the Material. Divine power tends to pool in the Outer Planes naturally—sometimes as planes, sometimes as celestials, fiends, etc., sometimes as gods—and sometimes as more nebulous things like whatever it is paladins tap into. (The Weave is probably involved, though how is anyone’s guess, since pre-5e the Weave was only in FR, and 5e doesn’t have a lot of detail on it.)
Even a paladin who never walks the planes is probably tapping into the beliefs of people from across myriad different worlds—and when one does planewalk, nothing much changes since that belief is everywhere. The published Oaths, at least, are presumed multiversal, i.e. sufficient belief exists to power them everywhere.
Overdeities can change things somewhat (probably). It’s probably within an overdeity’s power to just cut off extraplanar belief, meaning that any paladin who expects to get power from their oath better be sworn to something this world believes in. We don’t know of any worlds like that, and most likely Wizards of the Coast will never publish one because it might arbitrarily shaft some player characters and thus make players uninterested in going there (clerics, in particular, would have a very hard time, since the gods are almost-always extraplanar). But it seems plausible that such a world could exist, within the lore.
It’s also very likely that overdeities could choose to empower some oath that isn’t widely believed—on their world or any other. Whether or not that power reaches a paladin who planewalks, though, is pretty much up to the overdeity in question—they could allow the paladin to tap into that power even while on another plane. Or they could choose to block that. There’s no precedent for this, and at this point we’re basically talking about the DM-as-overdeity making up their own new campaign setting/premise; anything goes.
There are 17 Outer Planes despite there being only 9 alignments because there are Outer Planes “between” each of the non-true-neutral alignments, but these planes are still “aligned” even if, say, Pandemonium’s “chaotic-by-chaotic-evil” isn’t an actual alignment so much as a melding of CN and CE.
Prior to 5e, each region of the Material Plane—each campaign setting—was a “Crystal Sphere,” with the space between filled with phlogiston and plied by spelljammers. The recent Spelljammer: Adventures in Space book has ret-conned this by putting spelljammers in the Astral Plane—and offered zero explanation for how all of this works out. It’s unclear how the Material Plane is subdivided, whether there is any space between settings, or if there is any way to travel from one to another. (It’s also unclear why spelljammers are used in the Astral, which isn’t particularly dangerous or difficult to travel and has no need for the expense and complexity of a spelljammer.) But we’ve still got campaign settings, so the parts of the Material Plane are separate somehow. Or maybe there are many separate Material Planes—alternate Material Planes actually did show up in older material, but rarely.