Hi OP.
My DS1 is 7 now, but was exactly the same as your DD when he was a toddler. All my friends' children were confident and outgoing, and would toddle off by themselves at groups and softplay. I couldn't leave DS's side. He either spent the whole time sitting on my knee, or I had to be right next to him in the sandpit/ball pool etc. I also felt very isolated and wondered what I had done wrong!
I can't say exactly when he started tolerating social interactions a bit more (he also had the same response to children, if they tried to play with him) - but he is so much better than he used to be. He's not a social butterfly and is still quiet around strangers, but he's unrecognisable from how he was at 18 months.
I agree 100% with Eastie about it being helpful when people ignore him/don't make a big fuss etc. I have 2 sisters - one who would swoop on him with hugs and kisses and one who would wave and smile from across the room. Guess who he warmed too most quickly and guess who still makes him feel uncomfortable?!
As they get older, you find that strangers stop forcing interaction on them so much anyway (toddlers are super cute and everyone wants to chat to them!). So it does become less of an issue.
When the time came for him to start pre-school (I was dreading it!!) we spent a lot of time looking at photos of the staff on the school website, and even printed them off and stuck them on his bedroom wall. We then chatted to the photos (!), told them little jokes etc. So when he started, the faces were a bit familiar. He took a wee while to settle, but did in the end.
I also have DS2, who is the total opposite. He's 4. This time around I can happily sit and drink a cuppa with friends while he runs off and does his own thing. It's a totally different parenting experience!!