I don't know about the legality of adding the name except to say that most of the Catholics I know have done so. I come from a mixed RC/Anglican/Pres family.
But the name chosen isn't usually one chosen out of thin air. It is often (being careful here) one of the Saints, at least all the ones I know are. So the saint chosen is done by birthdate, child or family idnetifcation with a particular saint or even that the child resembles in character a particualr saint. In our family we have a couple of Saints - Ambrose and Joseph.
I note that Martha may come from St Martha
What is St. Martha patron saint of?
Saint Martha – One of Jesus' Closest Friends - International ...
The feast of St. Martha, a witness to service who confessed her faith in Jesus, is July 29. She is the patron saint of homemakers, cooks, domestic workers, waiters and waitresses, and hotel employees.
and
https://www.catholic.org/saints/saint.php?saint_id=79When I was confirmed into the Anglican church I received a certificate that recorded the day/date. Had this been confirmation been accompanied by a saint's name then I am sure that had I taken this plus my birth certificate somewhere to be verified it might end up with an 'aka' being recorded so eg Anna Frances
Catherine Brown where catherine is my confirmatory name I might be recorded as A F Brown aka AFC Brown
In days of old a person may be less worried about the idea of there being a problem with the birth certifcate and an added confirmation name.