Cis people have gender identity. If, as you say, cis people did not have a gender identity, that means that every gendered instance in all of society is explicitly done to inform others about the state of one's genitals. The wedding dresses cis women spend thousands on? They are to inform others around them that they have a vulva. The gender reveal parties? They are then about informing adult strangers whether or not your infant has a penis. If cis people lacked gender identity, everything from color-coding of infants to separated sports teams to separation of products by gender would be solely about informing others about other's genitals.
If cis people lacked gender identity, it would not make them uncomfortable to use the "wrong" gendered product when no one else could see. After all, if the gendered separation of society only exists to inform others about genital configuration, why not do whatever you'd like in private?
Mark, if you do not have a gender identity, I ask why you've likely avoided women's products for most of your life. Why not use women's deodorant? Or wear a blouse? Or use pink items? After all, what does it matter, these products are largely the same! Why does it bother you if a child with a penis plays with dolls, or has long hair, or uses nail polish? After all, they're just kids, it doesn't matter or impact their lives in any way.
These things matter to you because it makes you feel uncomfortable, in a deep, hard to place way. That discomfort is something conflicting with your gender identity, and is a small example of the dysphoria trans people experience.