My son & the mother of my grandchild were never married. They are no longer a couple but enjoy a very comfortable co-parenting relationship. When they broke up she and I did not! Although she was never my official daughter-in-law, that is the type of relationship we had and continue to have. Introducing her is always painful. She was never my daughter-in-law so she isn't my ex daughter-in-law - and even if she were that term is not acceptable to me -- she isn't my ex. The only phrases I am even partially comfortable with are "______'s Mom" and "the mother of my grandchild." It explains the blood-line, but it just doesn't capture the spirit of our bond. Are there words from other languages that might express the spirit of this relationship?
The OP asks about terms in other languages that might define her relationship with the mother of her grandchild. That is beyond the scope of ELU and my knowledge, but I can say with confidence that there is no English word or term for the OP's situation. English does not have an elaborate suite of words to describe complicated family relationships.
What English speakers (at least in much of the US) do is simplify, co-opt or make up terms to express not blood or legal relationships, but emotional relationships.
For example, my mother's best friend was Auntie Helen (later Aunt Helen) to me and everyone in our circle knew that. "Will your Auntie Helen be at your recital?"
As another example, a friend calls a woman who was close to him and his family when he was growing up "my other mother". I'm not sure of the actual relationship -- sister or cousin of one of his parents or maybe his mother's closest friend -- but it doesn't matter; she was his "other mother" and the listener would have to be dense indeed not to infer the love he felt for her, and her closeness to his entire family.
Thus, the OP can call this young woman whatever both feel comfortable with, and is not required to explain anything, although she may occasionally want to.
There's only one circumstance I can think of where the OP might have to back off from a title that implies a legal family relationship: if the young lady is in hospital and the docs are seeking permission for a medical procedure from a family member. Other than that, it is really no one's business but yours and the mother of your grandchild.