Your relationship with that socialisation was very different to mine. We both experienced similar pressures and expectations, and we both received similar privileges based on the way people perceived us, but our lived experience of that was very different.

An analogy is a closeted gay man that has to marry a woman and have kids due to social pressure. He receives many of the privileges of being straight, but it comes at the cost of himself. Calling that straight privilege erases that cost to him, and the price he had to pay by faking a role that wasn't his.

Nuance is important, and some privileges are real. But being forced to be perceived as someone that you aren't is not a privilege, and that shouldn't be ignored.