Well, since my child is only 9 months old, I wouldn't call myself overprotective, hahaha.
But, I am a little possessive over her, in that, I am picky about who can hold her and play with her, etc. I blame maternal instincts for it, though that may or may not be why I don't trust certain people. It's pretty common for first-time mothers to feel this way. It gets easier with following children, or so I've heard.
As a mother of a helpless infant, it is my job to protect her. Any time I think something may be wrong, it will either be instincts or fear. Most of the time it's fear. But there have been times when instincts led me to make good decisions to keep her from harm.
That's just the way I am. Every parent has a different story to tell. All parents want to protect their children. Being overprotective may not necessarily be healthy, but it certainly shows how much you love them.