getting one
Presents only drop while the Christmas event is running, and this is one of the rarer things hiding inside them, so expect to open a lot of boxes. Anyone farming presents for the weapons will pick it up eventually as a side effect. It does nothing in combat and takes no useful slot away from you.
the pet itself
The puppy trots along behind you on the ground and, when you get too far ahead or take to the air, spins its tail into a propeller and flies after you. That animation is most of the reason people keep it out. Halloween has the same idea in a different wrapper with the yarn that summons a kitten. The name is a real object too: a whistle pitched high enough that people cannot hear it but dogs can.