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.)