What term is used for a person whom you consider yours and don't want to share him with anyone?
It sounds as though you are describing a "thrall" or "slave". If the arrangement is mutually recognised and consensual then a "submissive", "pet", or "bottom" - taken to the extreme, possibly even "gimp". If the individual has been emotionally browbeaten into your thrall they may be said to be "whipped", with the vulgar "bitch" gaining increasing acceptance as a term of endearment.
If the arrangement is not mutually recognised and consensual , or if you are looking for a term like "boyfriend" or "husband", then eventually "ex", "victim" or "plaintiff" might be appropriate...
He or she is "spoken for".
If a person or thing is spoken for or has been spoken for, someone has claimed them or asked for them, so no-one else can have them.
I think the exact term strongly depends on who uses it. From the "owner's" perspective the term can be anything between
beloved one and
property, from the opposite standpoint it can be anything between
sadist. And a third party might describe it as
Something is definitely wrong with this couple or
These guys are really strange.
I'm going to assume that it's the word (not the person) that you consider "personal don't want to share".
So, a pet name:
A nickname, especially a name used by those in love with one another.
My boyfriend won't let me call him by his pet name, Pookie, in public.
Typically English speaking countries (America specifically) don't have any nouns related to ownership with no intent to share. The only example I can think of is if we are in a romantic relationship with the individual. Even then, "ownership" is implied rather indirectly. For example: "She is my lady". or "Back off, that is my girlfriend". If it is on a purely friendship level, ownership obviously wouldn't apply but jealousy may occur which is implied in a different manner.