I have an 8 year old daughter. We've been having the same experience you describe. My daughter is an only child, so I wondered if some of it has to do with not having siblings. She spends a lot of time with adults when not at school.
She is not shy and will approach kids she doesn't know, but I've noticed that as she gets older, she only does it if she gets a good vibe (e.g. a returned smile) from them. If she gets a 'look' (whether or not she interprets it correctly), she won't approach the other child. She is a very sensitive child... very kind/caring, very bubbly, etc. She doesn't seem to understand yet that not all kids are outgoing like she is and she shouldn't dismiss them because they aren't smiling and giggling, etc. like she does. She thinks they are mad or not interested in talking with her, which I told her is not always the case.
At school, she has a number of friends, but only one or two she considers "close friends." Her teachers have told us over the years that she gets along fabulously with all the kids in her class. When I mention my concerns about her being 'outside' the various circles, the teachers give me a funny look and say she is always playing with different groups of kids -- rarely sitting alone. It sounds like maybe she floats around among the various groups, and I think that's okay -- versus being tied to one group of kids. When I see her at school, she's always surrounded by other kids and seems happy. I thought maybe I was overreacting to her comments about feeling left out, but today while we were volunteering at an animal shelter, she asked if I saw the white dog in the playgroup and said that that is how she feels sometimes at school... being on the outside, wanting to join in with the other dogs, but the other dogs are ignoring the white dog. It broke my heart that she thought this from watching the dogs play. It came up randomly while sitting there watching the dogs.
Tonight I looked up meet up groups for only children and hope that might provide a way for her to get together with other kids outside of school... especially ones without siblings. I have thought about having kids over from school, but her closer friends live 40-50 minutes away, so it's a bit of a drive. Also my daughter has several autoimmune health issues and food allergies, so I do worry about how much some kids learn about her health issues in case they are used against her later at school so I'd rather she find friends outside of school to play with on the weekends. In a few years, if I remember correctly, there is a lot more girl bullying that starts to happen.
Good luck... I hope you find a solution that works for your daughter.