It's not that she lacks confidence, is just that she is a small toddler! She won't be like this forever and when she is 3 or 4 and really into running off on her own with her friends and tripping off to school everyday, you will look back and wonder why you worried.
My two were both like this at this age - terrified of hand dryers, chainsaws, noisy groups, separating from mummy.
Like many 2 yr olds your DD also has a new baby to adjust to.
Just give her time. As she develops over the next year you will see the separation anxiety and anxiety about her environment diminish as she gains experience and understanding about the world. Essentially from 2-3 yrs there is a shift so that anxiety about separation reduces and interest exploring the external world (new experiences, making friends) increases. With my youngest, the transition happened very quickly: at 2.5 he was clinging to my leg and I didn't reckon I'd have much chance of settling him in a preschool even by his 3rd birthday, but by 2.7, so only a couple of months, he was potty trained, talking 10 to the dozen and running off into preschool with his best friend without much of a backward glance at mummy.
In a year's time you will look back and know what I mean!
In the meantime, the best way to help her through this stage is reassurance. Her behaviour isn't problematic or a character failing, its completely reasonable, from the standpoint of a 23 month old - she doesn't know that there is no need to be scared of a loud noise, or that toddler groups are OK. She is at the very start of her journey of learning how to negotiate the world, what things she can control, what things she can't but that's OK, what things she can't control and are dangerous or scary.
Good luck xx
PS mine are still terrified of hand dryers at 5 and nearly 3!! But at least now I can just remind them that no-one is going to switch it on and it can't come on by itself!