My mother did/does something similar.

When ever I experienced an achievement it was never about how I had done well, it was always about how awesome she was for having created such a child or in some other way about her.

I don't know what damage this would do to a child specifically maybe some sort of self worth issues I guess. Like, they perceive that their achievement was really the parents and so they think they have never achieved anything themselves.

Boundaries between the two would probably be pretty blurred too. The parent kind of sees the child as an extension of themselves rather than an individual. (This is something I see a lot with my mother. She seems incapable of understanding that what's mine is not necessarily hers evidenced by her frequent theft of money, use of my things and invasion of my privacy.)