Background already is a thing in 5e. It has a particular design. So we can't really add to that and be compatible with 5e. However, we can create a new thing called "Culture" or something else that includes your background.
I am up for this approach.
If race splits off "culture" when becoming species, it is ok to merge the background design space into the new "culture" design space.
Essentially "culture" is "background".
But note. The Players Handbook background already includes: Ideal, Flaw, and what I call Quirk.
The "Ideal" is one sentence or so to convey a central concept. So here, I also have the player add a spiritual community that relates to this ideal. For example, here is the place where a cleric mentions a cosmic force or philosophy or polytheistic patron deity or pantheon, and a particular sacred community. As expansions of Ideal.
This is also where the player decides the Alignment of the character (such as Chaotic Good) and then adds a specific behavior that expresses this alignment (such as believing in personal redemption and being more likely to let a criminal go free without punishment, as long as future danger to others is less of a concern).
Likewise near Ideal, I add "Ambition", a longterm goal, usually spanning a tier. This goal correlates well with founding a stronghold and attracting followers at a higher level. It could be a wizard college with students, or a new spiritual community in a remote area, or a new import-export business, or so on. There may even be an ambition to become a leader of a region. Whatever the ambition, the player can already be thinking about the "stronghold" (whatever form it will take) from level 1, and achieve related goals toward it.
So these things − Alignment, Ideal, Spirituality, and Ambition are things that I do with the normal Background design space now.
If culture is moreorless the same thing Background is but maybe slightly beefier, such as adding proficiency with a weapon, armor, cantrip, minor ritual, or similar, that is fine with me.
Generally, a "cultural background" is worth about a feat of features.
So:
Background is often the job within a certain culture.
Ideal includes a spiritual community in the context of that culture. (But no extra mechanics: that would require a dedicated background.)
Region: the DM tells you where this culture is.