Social construct.

[The Grand Nudger]

But lots of domesticated animals become stray, and there's degrees of domestication. I don't think there's a strict precise definition of what's stray, feral, wild and so on.

There are definitions for all of those things (and genetic distinctions, too) - and yeah, there are degrees of domestication.  Dogs are fully domesticated, cats are not.  That's why stray cats are a bigger issue than stray dogs.  Cats, if left to their own devices, would not change their behaviors very much and have no trouble surviving and reproducing.  Our dogs (and our cattle, and our sheep, etc), on the other hand, don't last very long in a post human world.

I heard somewhere for example that pigs change quite rapidly without human contact and turn wild quite easily. I don't think a scientist would be able to observe this process and be able to state at precisely what time the animal became stray, feral or wild.
A spider would be a good example of how it can be my pet if I think it is and be a pest also if I think it is.

Domesticated pigs have trouble breeding true even with our help.  They are super smart, though, smarter than our own kids by some estimates, so they do alright when they get loose, considering.  Scientists can and do track feral populations, and this change does show up in their dna and behaviors.  

None of this has anything to do with social constructs, though, unless we'd like to contend that pigs and cats and dogs and weeds also create useful fictions in their own societies.  Something being a pet, a stray, feral, or wild....a weed, a domesticate, a semi domesticate...does represent an objective reality.  You could call a domesticated pig a wild hog but you'd just be calling it that, and vv, and the pig and the hog would continue to be distinct animals complete with a laundry list of genetic and behavioral disparities.  

OTOH, you could take a pair of human twins and put them in different societies, and they could be apprehended as a different "race" than each other.  Or... two very different people could just as easily be apprehended as being of the same race - This, because "race" does not represent genetics, or behavior, or in fact any objective reality.  It's a purely human overlay. There are lots of things like that - so you're bound to see plenty if you look. Probably has something to do with being a hyper social species in a vast and well connected society. However, there is more in the world, much more, than that which our social groups construct.