I can completely relate to you. My daughter is a year younger than yours, in pre-K, but she was exhibiting the polarity in behavior between home and school as well. One of the moms here on this site introduced me to the concept of Highly Sensitive People and Children researched by Elaine Aron - it has been a the key that has unlocked the mystery about myself and my daughter. I had a meeting with her teachers at school - they went out and bought one of the books: "The Highly Sensitive Child" and there has been a major, positive shift in my daughter's confidence and social activity. I urge you to check out the website www.hsperson.com and see if it speaks to you.
Is your daughter content in playing on her own or is she sad, anxious, or nervous? If she is content then work more on accepting her where she is. She may be picking up on your underlying energy of concern and slight frustration - if she internalizes that she may begin to think there is something wrong with her.
Playdates helped us. Her teachers strongly suggest getting together outside the class, so that the kids make a connection one-on-one and the confidence and new friendship carries over into the class. Take the initiative and invite people to your house or to a park for a playdate. Ask your daughter if there is anyone that she would like to meet for a playdate after school; or ask her teachers if they have noticed your daughter having some connection to one of the kids in class (maybe brief moments of smiling, laughing, playing, or holding hands) or a personality match.
With my daughter, the playdates with children of the same temperament don't always go well. Her most successful playdates are with kids that are more outgoing.
I stumbled upon something this past week at the parent/preschool open swim that we started going to...a girl about my daughters age swam over to me because I was holding about 5 little toys that I was using to play with my daughter. When the girl came over to me, my daughter was focused on something else and was about 5 feet away from me. I greeted the little girl, smiled and asked if she wanted to share the toys; she smiled and said, "yes" and we started having a little play exchange back and forth. What I didn't notice at the time was that my daughter witnessed that exchange - I modeled for her how you meet and greet new friends, and she saw the positive outcome. Quickly my daughter swam over to us and the two of them hit it off and spent the rest of the time playing together. It was exciting to watch her meet a new friend, and it dawned on me that I could have been modeling this step for her for the last year.
As far as whether to hold your daughter back in Kindergarten for another year - it may be a good idea, if she doesn't seem confident and content in making friends by the end of the school year. Look to your teachers they are the experts. Have you had a conference with them to discuss your daughters temperament - together you can create a strategy to support and nurture your daughter's development - you are a team.
Best wishes
M.