For example here are two personal scenarios I have dealt with.
1. I find a girl who is very physically attractive, has a lot in common , can have endless conversation with. I can talk to her about anything and she'll blow up my phone initially, beg for me to talk when I'm busy, fish for compliments, and generally show a lot of interest early on. While this behavior is overzealous, I didn't mind it as I actually liked the person and was thrilled she was that passionate about me.
2. I meet a girl who has a few common interests, somewhat easy to talk to but used misleading pictures and I ended up not being all that attracted to her in real life and once I saw more of her picture on facebook. This girl would also hit me up a lot, complain how I'm hard to hang out when I'm busy and told me how me going to concerts too much is the reason why we can't date even though when I humored her by asking her to do stuff, she said I live too far and get off too late. She also tagged me on facebook on our 2nd date when we weren't in a relationship and also publicly asked me out on facebook by tagging me in a comment. The fact that I didn't find her that attractive, and I'm not a shallow person, and her acting that way really turned me off.
Unfortunately, this can go both ways. I'm sure some girls probably thought I was clingy as I texted them twice after a date I thought was good and they didn't respond both times then quit contacting altogether.
Bottom line is I think clinginess is subjective.