I’m not sure what you mean by “who does Millie the dog do in Fallout Shelter.” If you mean the game character/pet, Millie doesn’t appear to be a standard named companion or pet in Fallout Shelter’s official character and pet lists; the game’s pet types are things like dogs, cats, and parrots, and the named legendary dwellers are a separate roster.

What I found

  • Fallout Shelter has legendary dwellers as named characters, and pets are a separate system.
  • I didn’t find any official Fallout Shelter entry for a dog named Millie in the material I checked.

Most likely meaning

  • If you saw “Millie” in a video, stream, or forum post, it may be the name of a real-life dog rather than an in-game Fallout Shelter character.
  • If you meant a specific quest, mod, or fan-made version, the exact reference would matter because it may not be part of the base game.

TL;DR

Millie does not look like a canon Fallout Shelter character or pet name in the official lists I found.