Is she actually unhappy though OP?
It's sometimes harder as a parent to see our child not behaving or living as we expected them to be than it is for them.
We're so conditioned in our thinking. We have all these ideas in our heads our everything should go.. Everything..work, relationships, marriage, pregnancy, newborn babies, toddlers... On it goes and it's not our faults, we're all just so conditioned and institutionalised. It supposed to be like this. And very often it isn't and leaves us questioning what we did wrong when in fact it's just life.
The more regimented we expect life to be, the more we struggle and panic when it doesn't work out that way.
I have 3DC, all very different. Eldest is much more social and finds friendships easier to maintain than the middle DC. Nothing wrong with middle DC, she is more like me and eldest is more like dad but already middle DD has moments she questions she isn't more like her elder sibling. I try to stop her right there and tell her there is absolutely nothing wrong with her, we're all different. She does have moments she wishes she had more girlfriends her age but then the rest of the time she is upbeat and telling us how kids her age just irritate the hell out of her!!
She is the same age as your dd OP and finds the company of adults preferable and we have encouraged that. She is a member of a sports club that is all ages.. She loves it and there's no forcing together of kids the same age. All ages train together. Also a competition hobby that is very much all ages, mainly with and competing against adults. And lastly a club which is focused on 10-24 year olds so a good mix of ages.
We are teens for a very short length of time. Why should friendships/social outlets be restricted to people the same age as us, doing what we expect a person of that age to be doing?! Give her opportunities to simply mix and find activities she enjoys, if she doesn't like it, she doesn't have to carry on. Try not to worry about friendships of the same age as long as she isn't or try to reframe her thinking if she is. Let's face it, how many friends do we all have as adults that are exactly the same age as us?! It's a bizarre way thinking if you think about it!