It is something that is bound to come with the feeling of love. It exists because of the basic need of humans to own and create a set of belongings
When one feels threatened by an external source the feeling of possessiveness comes into play. They want to make the external source aware of the fact that the object of affection belongs to them. This tendency is similar to the territorial tendency felt by lions, just expanded to many other aspects for humans.
Certain level of possessiveness is healthy but when it starts borderline obsession that is when one needs to understand his limits and draw a line. Control over ones feelings can help keep the level of possessiveness till a healthy extent.