Social construct.

[The Grand Nudger]

And I'm saying pets are a social construct. It depends on the outlook of what people agree on socially.

You just said to ask your wife about the spiders.
If I did and she said yeh they are our pets my husband is talking shit, and if you and your wife were to specifically chose to pay attention and care for one of those spiders and call it Sidney then that spider that isn't a pet would now be your pet spider.
In reality your wife doesn't want them as pets so they aren't.

You point out one difference between a pet spider and a wild spider, right there.  The care and feeding thereof.  Another would be their containment.  This changes their behavior.  People who work at zoos deal with this issue constantly, especially if they ever intend to reintroduce the animal.  In practice, this care and feeding also freezes them.  It's not always the case with novelty pets, which is to say non-domesticates...that they change, biologically, but the wild population still does.  A geneticist could, with enough samples, tell you roughly where and when an animal or it's descendants have been removed from a wild population (that's how we derive estimates for the domestication of dogs and cats).  Even in the exception seeking subset of wild caught animals kept as pets, you'll still find genetic divergence between wild populations and captive animals, changes to behavior, and objective differences in situations and circumstances.   

The word pet, and what we deem as "acceptable pets", a social construct.  The pets themselves, the fact that something is or is not a pet, not a social construct - because it refers to an objective reality about those animals.  A social construct is any concept that exists, not in objective reality, but as a result of human interaction. Compare the existence of pets, to the value of money. The value of money is not derived from the objective reality of a small piece of colored paper. The set of things that owe the totality of their existence to human agreement is large, and it's close to you (because you're a human). You see social constructs all around you - but, to continue along with the theme of pets, that's because you're in the fishbowl - not because everything is a social construct - or because the definition of the term is too vague. It takes aggressive misunderstandings to contend that a tree, for example, your own very first, is not a good example of a thing that is not a social construct. Every human being alive could blip out in an instant tomorrow, no one left to interact or agree....and there would still be trees. Our pets would linger on for awhile too, though, for most of them, not as long as trees.