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Find familiar allows wizards to send their familiars to a pocket dimension:
As an action, you can temporarily dismiss your familiar. It disappears into a pocket dimension where it awaits your summons.
Is it a demiplane or a extradimensional space?
If it is an extradimensional space, is it attached to the mage? Does it burst a bag of holding if the wizard goes inside?
There is no real distinction to be made between extradimensional spaces and demiplanes — they're just different words for the same thing. Per the Dungeon Master's Guide, page 68, “demiplane” is just what extradimensional spaces of notable size are conventionally called. You'll also note that DMG page 99 equivocates “demiplane” and “pocket dimension”, saying that either is large enough to serve as the exotic location for a dungeon, so there isn't a real line that can be drawn between those by size, either.
So both “demiplanes” and “pocket dimensions” are extradimensional spaces, and the difference between the two is more a matter of language being vague yet evocative: the two different words just lend nuances to describing the same thing, like how “boulder” and “rock” both mean a chunk of stone, just one suggesting greater size. The difference in the words “demiplane” and “pocket dimension” is just suggesting (vaguely) something about the size of an extradimensional space.
So given that one is just a synonym for the other — yes, a familiar's pocket dimension is an extradimensional space, and could be also called a (very small) demiplane if you wanted to.
No, the extradimensional space is not attached to the wizard in any sense. If it was, there would be very clear wording about the danger of the wizard doing things like entering portable holes or rope trick spaces. That's not something that would be accidentally left out of the description of find familiar. Given that lack, it's safe for both master and familiar to enter extradimensional spaces like demiplanes and pocket dimensions. Unlike a bag of holding, there is no portal attached to the familiar (nor is there rules text about the danger) — think of it instead as the familiar being “tuned” to that personal pocket dimension, so the magic of the spell can always “find” its home pocket dimension.
The text that SSD's answer refers to on p. 68 DMG says:
Demiplanes are extradimensional spaces that come into being by a variety of means and boast their own physical laws. Some are created by spells. Others exist naturally, as folds of reality pinched off from the rest of the multiverse. Theoretically, a plane shift spell can carry travelers to a demiplane, but the proper frequency required for the tuning fork would be extremely hard to acquire. The gate spell is more reliable, assuming the caster knows of the demiplane.
A demiplane can be as small as a single chamber or large enough to contain an entire realm. For example, a Mordenkainen 's magnificent mansion spell creates a demiplane consisting of a foyer with multiple adjoining rooms, while the land of Barovia (in the Ravenloft setting) exists entirely within a demiplane under the sway of its vampire lord, Strahd von Zarovich.
The the lower end of the size dimension range given for a Demiplane is as small as a single chamber, and that would be a lot larger than the extradimensional space needed to contain a tiny creature, or created by magic items like a bag of holding or a quiver of elhonna.
Also, you can use plane shift or a gate to travel to a known Demiplane. This would not work if the demiplane was tiny, too small to actually contain a small or medium-size spellcaster.
Not every extradimensional space counts as a demiplane, only those large enough. Mordenkainen's magnificient mansion, that is cited in the example, is large enough to be considered a Demiplane.
The pocket dimension of a familiar however has no defined size. It could be tiny too, or it could be a larger space allowing the familiars spirit to roam around. Depending on how large it was, it would just be an extradimensional space, or could count as a demiplane.
There is no danger of opening any planar rifts with it. The magic items that can cause such effects, such as a bag of holding or a portable hole spell this out explicitly. Find familiar has no such language. As spells only do what they say they do, there is no risk.
Demiplanes are extradimensional spaces that come into being by a variety of means and boast their own physical laws. Some are created by spells. Others exist naturally, as folds of reality pinched off from the rest of the multiverse. Theoretically, a plane shift spell can carry travelers to a demiplane, but the proper frequency required for the tuning fork would be extremely hard to acquire. The gate spell is more reliable, assuming the caster knows of the demiplane.
Emphasis mine, I believe the litmus test here is more importantly focused on than the size of the plane, but since this question was asked and answered no one seems to have focused on it. In this reading, for example, a familiar's pocket dimension would be an extradimensional space, but it would explicitly not be a demiplane, since it lacks any physical traits or laws in its description, regardless of how large (which also isn't described, making it appear to lack any physical qualities or laws based on a concept such as 'space' entirely) it is or isn't.
