Pet as a noun got some variety of meaning in English, so it's important which we take out of context.
1a: a pampered and usually spoiled child
1b: a person who is treated with unusual kindness or consideration :
2 : a domesticated animal kept for pleasure rather than
3 : a fit of peevishness, sulkiness, or anger —usually used in the
I think that "домашнее животное" ("domestic animal") is not really a good translation for the word "pet". It becomes even clearer that it is a somewhat different concept and a completely different idea when you remember how in English the word is also used as a verb and sometimes even as an adjective.
Also if for example you ...
There is the word питомец (etymologically, "the one being fed"), but
it has other meanings, it's less popular than домашнее животное, and, if anything, it's more bookish.
In compound words, it's sometimes replaced with the root зоо-: зоомагазин "pet store", зоотовары "pet supplies" etc.
Russian, as well as many other languages ...