Does your daughter know what "matching" means and that it's not an insult?
I've worked with a lot of children who,
when very little, get the wrong end of the
stick and assume a comment they don't understand is "mean" or bullying.
An example would be when a child (and much older than your dd, she was eight) came in on non-uniform day yonks ago with blue streaks in her hair, a sparkly top, bright leather jacket, leather look leggings and sparkly boots. My niece was the same age as her and wanted to BE Hannah Montana
so I commented "Wow, look at you Jessica, you are like a pop star! You look just like Hannah Montana!" with a big smile on her face. I caught her crying later and telling a TA I was mean but she wouldn't say why, she just kept repeating that I laughed at her and called her mean names but wouldn't say what. There was a complaint made against me and her parents came in all guns blazing.
She was sobbing and kept refusing to say how I'd upset her. In the end she managed to get out that I'd laughed at her in front of everyone, smiled "meanly" like I was making fun of her and said she looked "like a pop tart who was as fat as Montana".
She had never heard the term "pop star" and she was possibly the only 8 year old in the world who had not seen Hannah Montana and she'd decided my genuine, warm smile was a sarcastic piss take. Luckily, her sibling piped up..."erm no she didn't she said you looked like a STAR and were like HANNAH MONTANA. She was being NICE to you."
But for the rest of the year, even after we pulled up clips of Hannah Montana (and showed her she was even wearing Hannah Montana boots from the Disney Store) that child refused to see me as anything other than the "mean" teacher who humilated her and there was nothing I could do
to win her over.
And that's just one example. There are hundreds of child to child conflicts I can think of where the child just didn't understand or misheard ("he called me a CHICKEN", "no I didn't, I said I'm getting a KITTEN" etc). As the adults we often have to unpick what was really said and explain it in a child friendly way.
I.e "But sweetheart, it's not a mean thing to say someone is matching. It just means the colours you are wearing are the same and usually it's a compliment and it means the other person admires you. If she said that you DON'T match that would be quite mean but she didn't say that did she?"
Quite often little children get over excited and will pull one another over when they want go play with them. Staff try and intervene but again it's not necessarily a "mean" think, more an excited "COME AND PLAY WITH ME NOW!!!" thing. Ask her what a "mean game" is, what makes it "mean".
It sounds like she's just not used to play yet and interacting with other children. Bank phrases of "No Hope, I'm playing this game with Lily, now. You can play with us too and then, afterwards we can play your game"
might help.
I second the idea of arranging play dates with children other than Lily. Then you can also keep an eye on what behaviour really is mean and what might just boil down to misunderstanding. Are there any group Drama classes open near you? Sometimes these help kids understand the "rules of play" and also develop more confidence and assertiveness. Rainbows can also be good for this.