I've never been sure about this thing here. When a Guardian comes back in life, he remembers nothing about his own past. Always found this very intriguing but then... does a Guardian chose his own name after rebirth? Or does he remember it? I know that many Guardians have dreams about their past so, maybe, some of them could dream scenes of their past and ri-discover their names, but what for the others? And what about the Exos? Is the same thing for them?
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They know their names. In the grimoire it's shown that Guardians have dreams or visions about their past lives when they sleep or die. Its safe to assume that the name persists if those snippets remain. This is evident in the names of all Exos too. Even if their bodies were labeled with their name, they wouldn't know their number, which dictates how many times they had been wiped.