From this world logic and the ml's endless question, the name engraved will be their partners birth name (which is why the ml asked if mc change his name, cuz if his name is not his birth name then he's really not ml's partner). For the name twin, it's also definitely mentioned it's possible which is why mc is hellbent on searching another same name as him with same date. Well for that they most likely rely on clues (like when did the tattoo appears so they can guess their partner's age or birthday). Also, while having same name is normal but the partner's name is each other's name (like, if A have B on his body and B have A on his body then they're high chance a partner) so it's definitely harder to mistook your partner. That's probably the reason why "partner search" thingy exist in first place.
I hope whatever the fuck I wrote make sense since I read ts and wrote that explanation at 4am and is sleep deprived currently
What happens when you get a name change or have a name twin. There’s like a million John Smiths or Park Ji-hos.