There does seem to be a lot of importance on the idea of personal identity when it comes to Names. Part of learning something's Name, and having it be of any use, is also knowing how the person/creature themself pronounces it, so I'd say that if someone was born with one name, then adopted and given another, it'd be the name they grew up knowing as theirs that mattered.
I'd say that if a woman got married and took her husband's name, depending on her views of herself and her marriage, that could even cause her Name to change. I'm pretty sure, for example, that Charity's true name would include the Carpenter surname.