It's fiction. The writers say these characters came from the old Trek universe, therefore they did. There is no other existing universe for them to have come from, unless made up by the writers.
What visual differences there are, are so because of creative choices made by the producers - just like every other incarnation of Trek.
That is the explanation for every makeup and uniform change over the years.
As an aside: Spock Prime wouldn't have been termed "Prime"
unless he meant to have come from the original universe.