I don't have a gender identity. I know that I am a woman, ie an adult human female. But that's biology, not an identity.
A person absolutely can, if that's how you define gender identity. You don't have to agree with the definition though.
In your case, you don't think you have a gender identity and you are a woman, so by the defintion above (which you entirely dispute) your gender identity is "woman."
According to WHO, if there is no gender incongruity you are "cis," otherwise trans. Please don't kill the messenger when I say this!
A good analogy is neurotypical and neuroatypical. For example, if your proprioception isnt' work, you certainly notice it, if it doesn't you might not even be aware you have it.
Do you think gender dysphoria is a real condition, that is, people experience gender incongruity? If so, you acknowledge that most people don't experience it. Those people's gender identity matches their natal sex. Because they don't experience an incongruity, they don't "notice" having a gender identity.
The issue then is, does this mismatch mean that they are women, or is it like someone who thinks they are Napoleon when they can't possibly be? GC critical feminists say the latter.