Realize we are talking about a fantasy RPG here, and a person magically and divinely attuned to nature, animals and plants. We aren't talking Jeff Corwin here, we're talking Radagast from LotR, who can cast Animal Friendship and Speak with Animals at 1st level.
So yeah. I'd say that'd be pretty dang easy for him to do, honestly. If I were a DM, I'd make it a sidebar in a game, where they come across trappers that have an injured animal in a cage, and the druid needs to buy the animal from the trappers, or maybe some orc trappers that the party kills. The druid then needs to befriend the animal, heal the animal, and then train.
For balance purposes, I wouldn't let him get much advantage out of his pet though unless he judiciously uses Speak with Animals, Beast Sense, etc... But that's just me.
Also realize there's places that a wild animal won't go. A befriended wolf is still a wolf, and will really dislike going into a town or city. But others may be fine, like a small bird that you'd mentioned. Keep a nest under his hat, like Radagast.
