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.

No, familiars can be gained by means other than summoning

So there are three broad ways you can gain a familiar:

• By summoning one, using the find familiar spell;

• By forming a pact with a creature who will willingly be your familiar;

• By creating one yourself

In the first case, you — the player — have control over the actions of the familiar, as detailed in the spell description. But for the second case, the DM controls the familiar's actions, not you.

In other words, if you find a real pseudodragon in the game, tame it, and convince it to serve you, it becomes your familiar, but it is still an NPC. At any time, the DM may decide that you have violated the terms of your agreement/broken your mutual trust, and the familiar can leave you or betray you. It is choosing to be with you, but you don't get to decide its motivations.

It is also possible to have more than one familiar this way, if you summon one and form a pact with another one.

The Imp: the familiar that can be forged a pact with

In MM pg 69, you will find on the sidebar an "Imp Familiar" variant. They are described as:

Imps can be found in the service to mortal spellcasters, acting as advisors, spies, and familiars. An imp urges its master to acts of evil, knowing the mortal's soul is a prize the imp might ultimately claim. […]

Familiar. The imp can enter into a contract to serve another creature as a familiar, forming a telepathic bond with its willing master. […]

So there is the second way to get a familiar, as demonstrated in the RAW.

The Homunculus: the familiar that can be created

In MM pg 188, you will see the Homunculus: a tiny creature that is made by the wizard — not summoned or made a pact with. The text does not actually say the word "familiar," but its abilities as a companion are everything a familiar can do, and a little extra.

A homunculus is a construct that acts as an extension of its creator, with the two sharing thoughts, senses, and language through a mystical bond. A master can have only one homunculus at a time (attempts to create another one always fail), and when its master dies, the homunculus also dies.

Shared Mind. A homunculus knows everything its creator knows, including all the languages the creator can speak and read. Likewise, everything the construct senses is known to its master, even over great distances, provided both are on the same plane.