There's nothing in the rules that prevents you from having the service of two creatures with the Familiar variant.
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 one of these familiars. 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.
Needless to say, if even getting one familiar this way is entirely up to your GM, getting two is, even more so. There's nothing in the rules to prevent it, but you'll have to talk your GM into it if you want to have a quasit on each shoulder.
Now for combining Find Familiar with the Familiar variant: The interpretation that causes the least difficulty is that the variant Familiar isn't actually a familiar, it just "serves you as a familiar". In this case, there's no interaction between Find Familiar and the Familiar variant, and everything is fine.
However, if the variant Familiar is a familiar, well...things get weird. If you have your familiar from casting Find Familiar, and you then bond with one from the Familiar variant, there's no way to tell what happens. You "can't have more than one familiar at a time", so you've already put the game in a paradoxical state. Your original familiar might vanish, or your new one might die, or, well, anything, really.
If you have a familiar from the Familiar variant, and you cast Find Familiar, then by the rules (when using the interpretation that variant familiars still count as familiars), you get to change the form of your variant Familiar. You probably don't want to do this, since all the forms you could change it to are weaker than the one you've got, but there it is.
The fact that these rules break down completely when faced with each other is just more evidence - the Monster Manual is not meant as a player resource. Sorry, but it's not. Every spell or ability (like Wild Shape) that would require a player to look at the Monster Manual says "your DM has the stats" or something similar. The Familiar variant was never meant for players to see. It belongs to the GM.
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.
Best Answer
It's... complicated.
The truth is that, as you discovered, there is no data provided that canonically defines the creatures you can select as a familiar. Likely by design, the familiar is currently an undifferentiated bundle of stats relying on the player to flavor. This may change with the Gamemastery guide or Bestiary 2, but I wouldn't count on it. In other words, a familiar follows the template described in the familiars section regardless of what they are - they have your saves and AC, 5 * your level hit points, size tiny, low-light vision, and either a land speed or swim speed of 25 feet. Beyond that, they are an animal (further limited to tiny sized animals) unless otherwise noted by the feature that gives you the familiar (such as the Druid's leshy). That's the base, as you've noted, and the actual animal you choose is little more than set dressing.
I feel that the obvious intention of the designers was to not have one familiar be inherently better than another - "oh, I'll take the bat because it has blindsight and flight" or "I'll take the spider monkey because it has hands", and so on. So all familiars start off on this level playing field, and if you want your familiar to have flight or blindsight (not currently available) or opposable thumbs, you have to buy it using the familiar abilities. To this end, what the rule you are asking about is stating is that if you do happen to choose an animal that you (or a reasonable person) would expect to have one of these abilities, you have to take it, reducing the number of other abilities you can grant. Which, in a way, makes some animals inferior to others because you now have more limited options each day when selecting your familiar and master abilities. So, yeah, the implementation fails to support the intention, assuming my supposition is correct.
Realistically (with GM approval, of course), you could argue that your owl has a broken wing (or some Finding Nemo birth defect) that prohibits it from flying normally, so that if you don't want to spend a familiar ability on it one day to allow it to fly, it simply can't, and you aren't penalized for selecting a flying creature as a familiar (since the familiar abilities, as the artwork in the book calls out, allows you to put wings on a cat if you choose, so any familiar can fly).
In short, taking a familiar that naturally has any of the familiar abilities is a penalty by the rules as you are now locked in to spending those points, where it is optional for a normally less gifted animal, hence why I propose the hobbling of a creature's natural abilities as above so that you can have your owl for cosmetic reasons, but not be limited in your ability selection for it.
TL;DR
More to your question, you are correct that there are no stats for an owl. There are stats for a viper and an eagle (though it is small), but other than those, the closest that you can get to animal stats would be the animals described in the Animal Companion section, but they are size small (or med/large in the case of the horse) and are supposed to be considerably above average examples of their species, so it provides little more than an extreme upper bound. Here (and in the stat block for Eagle) a bird has a fly speed of 60', which you can't even get on a familiar if you get both Flier and Fast Movement (40' fly speed). So, it's far from perfect, and "bird" covers a lot of ground - it'd be fair to argue that owls have darkvision while hawks don't, and neither of them has blindsight, while bats should.
The best advice I can offer, aside from diving into some Pathfinder 1e books or waiting for more 2e books (or official online resources) to come out to clarify, is to just go with what makes sense and work it out wit your GM. Cats, owls, hawks, snakes (ignoring poison), and so on are pretty easy, but a bat, since there is not blindsight familiar ability, is a different beast (pun intended) entirely. Some would argue that blindsight might be too powerful for a familiar ability, others would say just house rule it in to the list, and others might suggest disallowing bats until official guidance is available.
Bottom line, it's a GM call, but I wouldn't sweat it since you have to pay for the feature anyway. It actually works out in the player's favor to say that owls don't have darkvision (or fly) and cats don't climb, so you can choose those abilities if you want, but aren't forced to.
An additional thought
The rules on familiars say that familiars can attack using your level as the modifier, but the lack of stats means we are on our own as far as determining the amount and type of damage.