Only you can answer this question after you have done a certain amount of introspective thinking.
Most people who are "possessive" are either (naturally insecure) meaning they can't believe someone could "actually love" and want only them or they don't trust themselves to (choose) quality mates for themselves. The latter is especially true if they've had past relationships where they were cheated on or betrayed in some manner.
The betrayal may have even come from family members who have proven to be untrustworthy or undependable. Naturally if a person can't count on the love of their own family members then it will be extremely difficult for them to believe an "outsider" can be trusted to be loving and completely loyal to them.
People who don't believe "good things" happen to them are never comfortable when it appears they have a stroke of good luck. They're always waiting for the "other shoe" to drop or have the bottom fall out.
This causes them to "keep an eye" on their mates. Some even resort to snooping through cell phones and emails. Others put their mate in the "hot seat" demanding they (prove) their love and loyalty over and over again even though they will never be satisfied.