What a tangled situation. You characterize the woman’s behavior as competition with you. I think it is far more complicated than that. On the one hand, she decided to be a rather absent mother and made no objection to you doing things that a mother usually would for her daughter. If she struggles with mental illness, she may have thought it was the best way to love her daughter at the time. On the other hand, she may feel some sadness, guilt or shame about having done so. I’m guessing that maybe she is trying to make up for lost time with her own daughter by planning the vacation. She may also be attempting to balance her daughter’s relationship with you by attempting to be closer to your daughter.
It makes sense to me as well that the other girl wants to visit you with your daughter and does things like opening the mail you send. After having been a part of your family for years, she may not be able to understand or respect a new boundary.
This may be going on at a very unconscious level. It could well be that neither the mother nor the daughter are being purposeful in what looks like hurtful behavior. They may not know how to have a more usual and closer mother-daughter relationship when you’ve been in the mothering role for so long. The daughter may not know how to ask you to continue being a surrogate mother without feeling like she is betraying her biological mom.
One challenge now is for you to find a way to move out of the mothering role without making the girl feel abandoned. Another challenge is to not take offense at the other woman’s efforts to “adopt” your daughter. Arguing with her will only stress you and it is unlikely to change her.
Your daughter knows who her mother is. It’s fine to remind your daughter that it isn’t helpful to recruit the other mother into any disagreements the two of you have. Shifting your relationship to a more adult — adult one now that she is a young adult means working things out directly with each other, not pulling someone else into problems.
Because this is so complicated, you might find it helpful to talk to a family therapist about how you can shift your relationship with the other family and affirm your relationship with your own daughter. An experienced family therapist will be able to give more informed guidance and support than I can.
I wish you well.