The explanation for a Doppelganger's shapechanging is rather modest and leaves a lot to be explained. In a recent homebrew game, a player was playing a Doppelganger as a playable race, and shapechanged into an aquatic race, claiming he could now move with a Swim speed and breathe underwater. Is this accurate?
The doppleganger's shapechanger ability specifically states:
Its statistics, other than its size, are the same in each form.
Meaning that it keeps all of the stats listed in the doppleganger stat block (other than its size) and does not gain any stats from the form it changes into.
I am not aware of any official doppleganger playable race though so if this is also a homebrew then the decision ultimately falls on the DM.