But why would he need to be the same person? (I think Ivan and Natasha just named their two sons the other way around, and Pyotr got the approximate body originally reserved for Pavel. And again got killed by Klingons. The bastards!)
Anybody younger than Kirk is free game for being genetically different, which is why the more radically different looks of, say, Uhura and Sulu don't bother me much. Might be their sexual identities are genetically predisposed to be different, too. Or their tastes when it comes to choosing a profession within Starfleet (as often discussed, linguistics appear a Kelvin Uhura phenomenon exclusively, while physics did not seem to interest the Kelvin Sulu much).
Timo Saloniemi
"Chekov doesn't have a brother"
- Sulu, "Day Of The Dove".
If all the characters post Kirk are genetically different then we're not actually seeing Kelvin Counterparts of the Prime crew. That defeats the entire premise of the Kelvin movies.