No, I don’t think the Brown court would have said that people could be separated by race on a train. Brown fought against segregation in schools because it was unconstitutional and violated the 14th Amendment. I can’t see why they would agree with segregation in other institutions.
The Plessy case, which occurred quite a bit earlier than the Brown case, ruled that segregation was okay as long as it was “separate but equal.” If it was Plessy v. Brown, Brown could have made the argument that “If it has to be separate, it’s not truly equal.” There’s no reason to divide the people unless there’s unfair treatment involved.
The main way in which the colonies developed identities independent of great Britain is that they soon were home to a slew of immigrants from around the world, who brought with them their own traditions.