It can. But doesn't necessarily.
I am submissive to one partner, Dominant to the other.
My submissive partner has a tattoo on the back of his neck acknowledging his submission to me.
We do have a rule:
No one else Dom's my sub and I have no other subs..
No one else Dom's me but my Dom and my Dom takes no other subs.
It's ok to play with BDSM activities with others if one wishes-but no formal recognition of position.