My mother has a Mazda CX9 with a Bose subwoofer in it. Would I be able to take it out and put it in a VW passat? Answered

Yeah. I think that would work.

But you have to ask your mom first. ;-)

Bose is known for high end audio, a subwoofer does not fall into the high end category, especiall not with the audible limits of a car.
Just for that reason I fail to see why ripping one out of mums car is better / easier than investing a few bucks in your own...

In principle yes. In practice many things will get in your way. you may or may not have the skills to over come them.