Byleth getting an S support will obviously lead to some kind of event and possibly affect the story etc. Basically it matters during the game. Maybe the others' A supports can then work like in the GBA games, meaning some of them will become couples after the story ends. I know this has been suggested already, but I'm just saying it would make sense having different ranks for that reason.
If it's true that you can have multiple A supports, this may speak against it. Though there could be just one canon partner and you need to unlock that particular support to unlock the ending, while other A supports are of the friendly kind.
I actually don't believe this is how it's going to be, but it could.
I've always liked data transfer between games, having my actions in one game matter in another. So what I'd be interested in would be supports having an effect in a potential sequel. I wouldn't bet on this either though, but we did have something similar between FE9 and FE10. But I just haven't seen data transfer being utilized between any games for a long time.