Also having the same issue so would love to know the answer.
I (xbox x/s) trying to play with my friend (PS5). I'm unable to find him when I enter his tag into the search bar. He found me by my xbox gamer tag and added. Accepted request but there is no way to invite him to a lobby and vice versa. Checked crossplay settings on both ends. I did realize that my EA ID is different from my XBOX Tag. He found me by my EA ID as well but it says that I'm "offline" and my xbox gamer tag is "online." Cannot figure it out for the life of me but it's getting real annoying. Appreciate any info or if anyone found a fix for this.