The Dr originally wasn't autonomous, it could be argued he's just part of the ship, but the holo emitter changed that.

There's an argument to be made that that changed the moment he started to be established as a sapient individual of his own.

I'm amazed the Daystrom institute let him keep it, but since it's apparently his, and that makes him autonomous, I would argue he's just like Data (minus the permanent corporeality of course).

It bring future Federation technology bequeathed to him may help there too. The Federation likely doesn't want to risk issues with the Time Police by confiscating and studying the emitter, so just let him keep it to do with as he wants.

There's also an ethical argument that removing it would severely restrict his ability to move, given that Starfleet would have trouble furnishing him with a sufficient replacement.

I suppose there's a question about ownership given his origins as a Starfleet asset, but since he can be replaced with a copy of the original program, there's no real material loss in letting him leave the ship.

We also know from Prodigy that the Voyager was intended to be shelved for study, so it no longer being active might also be a good reason to allow the Doctor to roam about, instead of effectively trapping him on the inactive ship while Starfleet scientists pulled it apart and studied every crook and nanny.