Wizards can gain familiars without find familiar and can have other familiars, at DM's discretion
For wizards there are three ways to obtain a familiar: (1) via the find familiar spell, (2) by adventuring and meeting a creature that itself has the ability to become a familiar and is willing to do so, (3) by meeting a creature that they can bond with as their familiar by casting find familiar on it.
All but the first one require your DM's buy-in to create more colorful wizard characters. The main concern there is that using other creatures may be a power play or unbalancing, so be prepared for the DM to say no to more exotic options for your PC. Having a tree-hugging Elf Wizard with a Pixie familiar may be flavorful, but also would be entirely overpowered at low levels.
Summoning with Find Familiar
In this case the familiar is a fey, celestial or fiend spirit that takes one of the beast forms described in the spell, and has the abilities described in the spell.
In addition to the beasts in the spell, various other creatures have been listed in official adventures as options for summoning with the spell if the DM agrees to it: a Tressym, Almiraj, Flying Monkey, Abyssal Chicken, Fox, and Hare.
Strixhaven: A Curriculum of Chaos has a feat called Strixhaven Mascot (preconditions are 4th level and another feat, Strixhaven Initiate), that allows you to chose the following option as your familiar, depending on your college: spirit statue mascot, art elemental mascot, fractal mascot, inkling mascot or pest mascot.
While these are the ones listed in published materials, talk to your DM if you want another familiar. If it is just for flavor, reskinning the statistics of an existing one should be unproblematic (a toad with frog stats, a hamster with weasel stats etc).
As mentioned in the NPC section in the MM (p. 374) and in Volo's Guide to Monsters (p. 213), for NPC spellcasters that can cast find familiar any Tiny monster might be an option, such as a Crawling Claw or Cranium Rat.
Creatures opting to act as familiars
The Monster Manual lists a number of creatures that can act as familiars. These monsters include: the pseudodragon, quasit, and imp. Volo's Guide to Monsters adds the gazer (which requires a 3rd level or higher spellcaster to bond with).
These monsters serve as familiars of their own accord, and can end the relationship at any point if they so chose. This means it is up to the DM to decide if one of them will be your familiar. They are not limited to wizards as companions.
They differ from the familiars summoned by find familiar, in that they are actual creatures that will die (or be sent back to their home plane) when killed. Find familiar does not impose any limitations on them or grant you any abilities for them. For example, they can attack in combat and you cannot send them to a pocket dimension or deliver touch spells with them. The bonding abilities you enjoy when they are your familiar are granted by them, not the spell. They include:
a telepathic bond with its willing master. While the two are bonded, the master can sense what the familiar senses as long as they are within 1 mile of each other
while they are within 10 feet of their companion, the companion shares their Magic Resistance trait [for pseudodragon, quasit, and imp]
Bonding creatures with find familiar
This is a mix of the other two options. The familiar in this case is a creature encountered adventuring, but the adventure requires the wizard to cast find familiar to bond with them.
In both cases the adventure text does not elaborate in detail how the bond works once established, and this will be up to the DM.
In our case, the DM ruled for the gazer that it behaves as described above, the range for the telepathic bond is 1 mile, none of the abilities or restrictions of find familiar apply, except that it has no freedom to leave the bond. (Our DM also allows casting find familiar on other Tiny creatures to bond them, but reserves the right to veto -- no Fairy Dragon familiar, for example.)