My town thinks I'm that person! Twice, people have brought over stray dogs that got hit by a car. One lady tried to re-home a goat with me, and got rather difficult over my refusal. I'll take unwanted chickens if the owner is OK that it is getting butchered. Otherwise, I always say No (we have three shelter dogs already), but one neighbour talked me into taking in a baby wild rabbit. (Except, that it wasn't a rabbit, and I still don't know the creature was, and it unfortunately died a week later. Both to my sadness, and relief - it had huge alien eyes, but only a stub for a tail - reminded me of one of the Pokemon characters.)
Animals also find me. Am currently caring for what is probably a baby Ringneck (Collard ?) dove. Found it on the ground, but couldn't find the nest. We've had a lot of storms recently.