part of identity is our relationships to other people.

I wouldn't agree, simply because I consider relationships as existing between people, not within them individually, and more as 'facts of the matter', as opposed to immutable aspects of individuals themselves. But again, this is simply a disagreement on the definition of "identity". I'm not saying your definition is wrong, but it obviously is different.

A familial connection is a fact about someone's lineage, but it is no more a part of someone's identity than to the extent that that individual chooses to make it so. If I was adopted and have never met the woman who birthed me, then yes, she's still my mother even though I never knew her. But that being a fact has no inherent relationship to my identity. The same is true if I was raised by my birth mother but am now estranged, and she has no part of/in my life—she'll always literally be my mother, but in this case, her existence is no part of my identity any longer.

Nonconsensual trauma that alters one's sense of self against one's will is the only thing that muddies this water at all, I think, but even in a case like that, it is only from within that whatever degree (whether zero or nonzero) those events shape one's identity, can change.