> don't think those things can exist apart from religion (at a cultural level, anyway), because as soon as you decide morality is relative then everyone is free to choose the morality that is most convenient for them/their tribe.

You've confused absence of moral relativism with religion; the former does not require the latter, and most non-religious people are not moral relativists, at least in the sense of the strong meta-ethical relativism which would justify the claim above.