I believe mage hand was first codified in 3rd edition; prior to that, it would have fallen under the purview of the unseen servant spell or the cantrip spell.
In 3rd edition (and 3.5), Mage Hand is nothing like summoning a spectral hand; it was merely telekinesis:
You point your finger at an object and can lift it and move it at will from a distance.
In 4th Edition, the spell changed significantly, becoming a conjuration, and now there's an actual hand involved:
At-Will * Arcane, Conjuration
Effect: You conjure a spectral, floating hand in an unoccupied square within range. The hand picks up, moves, or manipulates an adjacent object weighing 20 pounds or less...
Conjuration is described in 4th Edition in this way:
Powers that have the conjuration keyword create objects or creatures of magical energy.
So as far as 4th Edition is concerned, there is no specific information about what the hand actually is other than "magical energy". It's not stated to be an astral form, left or right, or anything else.
5th Edition largely keeps the design of mage hand as it was, and remains vague about its nature:
A spectral, floating hand appears at a point you choose within range.
You can use your action to control the hand. You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial. You can move the hand up to 30 feet each time you use it. The hand can't attack, activate magic items, or carry more than 10 pounds.
The rules about what a conjuration is don't really help:
Conjuration spells involve the transportation of objects and creatures from one location to another. Some spells summon creatures or objects to the caster's side, whereas others allow the caster to teleport to another location. Some
conjurations create objects or effects out of nothing.
As far as the main player text goes, it's a magical construct but other than that, there's no specific information about what a mage hand "really" is.
Tasha's Cauldron of Everything has a section on Personalizing Your Spells (p. 116) that discusses altering the imagery associated with a spell fit a theme or personal preference. I won't quote the entire section here, but suffice to say that minor details of spellcasting such as what a mage hand looks like are entirely up to you to decide. If you want your mage hand to manifest as a spectral tentacle reaching out of a hole in space to manipulate items, a ghostly skeleton that moves objects at your command, or an incorporeal crab that flies around to do your bidding, that's perfectly fine; just realize that it doesn't change anything about how the spell functions. I would assume the mage hand should remain spectral regardless of its form, to reflect the fact that the hand (or whatever it is) can't be attacked or interacted with.
Since the book is silent on what the hand actually is, it should be up to you and your DM to make those kind of decisions. Depending on what kind of caster you are, if you prefer to have your mage hand be a ghost or extradimensional entity that you're commanding or a fragment of your astral form that you're projecting out of your body, there's no real problem with it. It won't change what kind of spell it is -- it doesn't become necromancy just because you want it to look like a spirit doing your bidding -- but as Tasha's says, spells are meant to be customized to say something about your spellcaster, if you so desire.
There's no game effect to saying "this is actually the tentacle of my pact patron's servitor" or "this is a friendly nature spirit that I'm empowering" or "this is the astral form of my left hand". That's up to you. There isn't a single solid lore answer to what a mage hand is; it is what you want it to be.