Ownership is expressed in many ways to different degrees.
Sometimes it is only indirectly implied; while at other times it is much more omnipresent.
The use of various things from: abstract objects as mundane as a clothespin or ribbon, even a picture, to the collar or a brand or a wedding ring, can be used in identifing one as property or as having a mutal covenant of dominion.
All help, in reinforcing a submissive's focus on her dominant.
Many dominant's wish for thier girl's first thoughts and primary focus to be about him; just as thier own thoughts should be with great consideration upon her.