Well seeing as there are no Guardians named Terry, Mohamed, Alison, or Pedro, or any other common name, I'd wager that they might remember some aspect of their birth given name, but they pick their own name later on.
Unless culture and names have changed drastically in the relatively short 300 year span of the golden age. Seriously, who would name their kid Shaxx? Or Felwinter?
English
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My mom was named Shaxx. We don't talk about what she does for a living.