Gender identity doesn’t get assigned at birth. There is no “gender” field on a birth certificate.

Sex gets identified at birth (at the latest, usually before, during pregnancy, unless specifically requested to keep it secret).

Two reasons this is important to point out:

• Assignment implies that the act of assigning is what makes it so. It’s not. If a doctor says that a male baby is female, it’s not now female just because they said so. “Identify” is a much more accurate description of what the doctor is doing.
• The whole premise of “transness” being a thing relies on the notion of sex and gender being two distinct, independently-variable traits. So be careful not to conflate them. It causes needless confusion, since conflating them literally undermines the whole thing–after all, if “sex” and “gender” are equivalent, then it’s objectively impossible to be trans.