S.H.
She sounds very normal.
She is a little kid.
This is common.
What can you do?
1) Don't compare her to other kids.
2) Let her be herself... teach her that she is fine, and to know herself and her own cues. I taught that to my kids since they were 2 years old and up. Thereby, a child will KNOW themselves, and their own cues, and be self-assured. And it enables them, to follow their instincts. Not just being followers.
3) Both my kids were "shy" at that age too. So what. No biggie. I let them be themselves, and taught them to tell me their feelings. And I never made them feel awkward about it. And they became very self-assured individuals. Who trust themselves. That is key.
4) 3 is a very young age. Your daughter is fine. Don't worry. Nurture HER... and for who SHE is. That way, she will be her own person. And that is very important, as a child grows up. So that they know who they are and will know their own cues.
Kids this age, do not necessarily fully engage and interact with other kids. They do not socialize like adults. They don't even know the full spectrum of socialization yet. Their "emotions" are not even fully developed yet, at this age. So teach her the names for feelings and that she can tell you, anytime, grumpy or tired or happy or sad. Then she will not feel awkward about it.
All kids are clingy or needy at times. No biggie. They are young. They need us to guide them.
And not all kids are extroverts. Being extroverted does NOT make one kid better than the other. There are MANY great successful people... who are shy or more introverted. Extroversion, is not the only way to be.
Let your child, be herself.
And teach her to know herself and her own interests and cues.
That way a child will be successful and trust themselves. And be confident.
"Shyness"... does not mean a child is less confident.
So please don't confuse that.
My kids were shy and still are shy with certain things. BUT, they are very confident kids and they KNOW themselves. They have good ability to discern things.
Don't worry.
My kids still napped at 3 years old too.
My son is 5 now, and he STILL naps.
Having to nap... is not a "bad" thing.
A child needs to nap, when they need to nap.
And yes, not napping makes any kid cranky or fussy.