The Find Familiar spell is currently the only method for a wizard to obtain a familiar, and it lists the creatures available. However, a large number of creatures in the Monster Manual, such as the pseudodragon, imp, or quasit, include the 'Familiar' variant. Note that a Warlock with the Pact of the Chain feature can obtain many of these creatures as a familiar.
What you have to remember is that variant monsters, like monsters, are designed for the GM to use to make enemies more interesting. The Familiar variant is a monster variant, just like the troll's Loathsome Limbs variant or the Genie Powers variant. They're for GM use rather than player use.
The Mage NPC in Appendix B of the Monster Manual also has a Familiar variant, which says:
Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell, (see the Player's Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.
So the Familiar variant is for GMs to create more interesting NPCs, rather than to provide players with additional options. Of course, with your GM's permission, you could obtain a more interesting familiar. This would probably involve actually finding such a creature and somehow forming a bond with it. But this relies solely on your GM to allow and arbitrate.
It's also worth considering that allowing a wizard to obtain one of these more powerful creatures somewhat invalidates the Warlock's Pact of the Chain.
First, as Derek Stucki mentioned in the comments, the errata for the Ranger's companion animal clear up some of this:
Ranger’s Companion (p. 93). Like any
creature, the beast can spend Hit Dice
during a short rest. If you are incapacitated
or absent, the beast acts on its own,
focusing on protecting you and itself. It
never requires your command to use its
reaction, such as when making an opportunity
attack.
Out of game, the rest of the discrepancy is about balance. The Wizard's familiar cannot attack, but the animal companion can and often does. Allowing the Ranger's companion animal to attack independently essentially gives the Ranger an extra Attack action each turn, which would break the action economy.
This Q/A addresses the issue of balance if a Ranger could give their companion a single command to Attack and then have the beast continue attacking without giving up any further actions of their own. It concludes that it would indeed be overpowered.
In game, we can understand the discrepancy when we understand that a Familiar is not merely a beast, it is a spirit (celestial, fey or fiend) in the form of a beast.
Best Answer
Nothing.
There is no current rule that spell effects end when a caster is unconscious or dies unless they are specifically concentration spells.
Thus, since Find Familiar is not a concentration spell, it's perfectly reasonable to assume that the familiar stays and is unaffected by the fact that his wizard is unconscious (or dead).
And in fact, the duration of the Find Familiar spell is "instantaneous" meaning that once the familiar is created, it's there, and there is no continuing effect of the spell. It can't even be dispelled.
It's important to note that if your DM wants to rule that current spell effects (like say, spiritual weapon), end when you go unconscious, it would not affect a spell like Find Familiar since again, the duration of that spell is instantaneous.