Translation:This tiger is sad, because it lives in the zoo.
I would say it's something like this:
Mieszkam w tym domu. = I live (reside; my home is) in this house.
Tygrys żyje w zoo. = The tiger lives [out its days] in the zoo.
Mieszkać I believe always refers to having a home, whereas żyć can mean what it does in this example, or it can mean 'to be alive'.
(In Finnish we have two separate verbs for 'live', just like Polish does; English can use the same verb for both, which makes it tricky.)
A British native just told me that "it" is really fine for animals and while we are perfectly aware of the usage of singular "they", we do not accept it because there is a risk that a person using it did not do it consciously but rather misunderstood the Polish sentence/made a grammar mistake in English. And Polish is so gendered that it also serves as a way of showing it.
Wouldn't it actually depend on the gender? “Ten” is the male grammatical gender, which applies to the man as well as the tiger. Hence, I wouldn't say that applying any non-neutral gender on animals indicates a closer relation to the animal, but rather human names for animals, for example.