In the end of Alien: Covenant (2017), David reports to Mother and she seems to recognize him (she even calls him by his name). How does that happen and David is not treated as a foreign entity by her since he was never a member of the Covenant crew?
Because David also belongs to the company.
David belonged to the Weyland corporation, just like Mother and Walter. It is likely that Mother had a mainframe with all associated personnel of the company so that any officers could work with any vessel.
As such, David still has his credentials and authority in the ship and utilizes them without any hassle.
If you, however, ask me how he knew Walter's credentials at the end of the movie, then that is something I can't answer...