Hi - just my 2 cents, but my son was very similar and my advice would be, don't push it. It will just make you and your daughter more stressed, and in reality, there is probably nothing at all wrong. I was also concerned that my son was extremely clingy, not well able to interact with other kids, freaked out if another kid got on the slide etc. Tried to gently encourage him, but it often ended up just backfiring or at best, having no effect. He started growing out of it around 2 1/2 and now, at 3 1/2, is much better. He is still not the most outgoing child, and still won't participate in group singing, but he's well able to play with others and to enjoy his school activities in general.
We found pre-school helped a lot (he started at about 2 3/4). You may find it takes a longer settling-in period, but once established, the familiarity of the routine there can really make it easier. If you have any choice at all, your DD would probably cope much better with, as someone here said, a "sedate" pre-school environment (we do Montessori, which some people hate because it's strict, but it works well for us). My son still cannot deal when there are lots of screaming kids in a room, or in random free play situations. The structure of his pre-school has helped make him comfortable there on his own, as far as I can make out.
Hope this helps. Don't despair, just try not to over-think it all. I know how frustrated it can make you feel to be in this situation, believe me! But it's not anything you've done, nor is it likely to be anything you can really control - kids are just different. And she will definitely grow out of it, but not till she's good and ready