Interesting— I usually think about this as a Catch-22 insofar as a cis person asks a trans person: "What qualifies you as a member of your gender?" Which can only mean: "To which gender-typical behaviors do you conform, or which gender-typical features do you manifest?" (And the stakes, of course, are whether they're able/willing to recognize our gender.) So the trans person lists a few things that make us "a woman" or "a man" in a babyish 101 kind of way, to establish some basic level of understanding and recognition. The cis person who is entry-level on trans issues may hear "I must be a woman because XYZ womanish things I do." Unfortunately, the trap has been laid, as later, down the road another cis person will say: "Ah ha, so you trans people base your identities off babyish, regressive stereotypes?" It's a double-bind, because if the trans person says off the bat, "Thanks, but I don't conform to any gender stereotypes," the response will be, "Well, then, there's no reason for me to recognize you as one gender rather than another, so you can just stay what you were assigned at birth, due to my inability to understand your gender and because I say so."

You're pointing out something important — maybe there's a couple different meanings of stereotypes at play here. One is simply a statistical observation of whether a feature is "typical" of a gender (e.g., nail polish is usually more of a girl thing than a boy thing), and another is whether you're reduced to that feature in some sort of causal relationship and thereby objectified and dehumanized (e.g., I am nothing but my nail polish).