When I took my cat through a nether portal that led to the overworld, it did not come out the other end. The portal it was to exit from was a very large one and I assume that is why this had not happened with normal sized portals. I did some looking around and went to the coordinates where the portal was in the nether, but in the overworld and it was there.

TDLR: Cat teleported to its nether coordinates to the same coordinates in the overworld when going through a portal.