I wouldn't worry too much purely based on "School say she is happy and friends with everybody st school". A teacher would have nothing to gain by pretending everything's rosy when your daughter's struggling.
Some children do float around friendship groups at that age. Maybe it's the kind of adult they'll become (and nothing wrong with that) and some because they haven't matured as much as others in Year 1. They're still babies really and depending on their socialisation (attending school, siblings etc) as much as anything else, they can easily struggle initiating play, joining in games and how friendships work.
You say she isn't "shary". You mean she doesn't like to share. That's an important thing and should be addressed. It will quickly put other children off playing with her as professionals and parents put so much time into talking about sharing, give-and-take and all that it entails at this age. Try simply talking to her about it. Encourage compassion and empathy - something Year 1's tend to be developing at this age.
Why do you think she's unhappy during play dates? Have you been the host as well as the guest? Is her reaction the same?
Some of it can be down to school gate politics. Not specifically excluding others but if a group of parents are friends it's likely that their children will spend time together out of school. Find out if your son has anyone he would like to play with more and introduce yourself to their parents. Make an effort on their behalf. I've been in the situation where my children's friend's parents were wankers but needed to suck it up*
Do voice your concerns, email to the teacher may be best. Explain how your daughter's unhappy. Someone'll keep an eye on her during break / lunch, encourage play and either consider strategies or tell you you're being "overprotective and a bit hysterical".
*not my strong suit
(ex-primary teacher, now head),