I have 3 children aged 23 months, 5 years, and 10 years. I've found that when they start walking and gain independence, they often become more clingy and fearful for a little while. When they were held more, it was easier to face many challenges. It is ok to still need that closeness and the reassurance that comes with it, even though the other kids seem to be able to join in right away. It can be scary for a small person to enter a group of people. Also, some kids are slower to warm up and aren't ready to be approached and join in as fast as others. There are books and theories about the different temperment styles...
My oldest was clingy, fearful and slow to warm-up. I worried too. She still has these traits in her personality today at 10 years. My 5 year old went through a similar stage, but for her it was simply a stage she grew out of. She eventually became outgoing and very friendly, a real people person. Now, with my 23 month old, I'm seeing him go through a similar stage. This time, I'm able to relax and let him become whom ever he becomes without worrying so much. I hope he becomes less anxious and more outgoing, because I think life will be easier that way. But, I've learned from raising my others that it could be a passing developmental stage or it could be personality. Either way, they need reassurance at this time.
It is hard to watch when you want her to love life and be social and friendly. I still have moments when I struggle and feel sad for my oldest daughter. The best you can do for your daughter is to not get anxious about her while she is anxious. If you are calm and give her space to slowly warm up to people, she'll learn eventually how to do this. You sound like a very caring and devoted mom. Lots of luck!