I'm not going to give my seat up for a lady
Is that because you do not abide by that societal norm?
Exactly, I don't care about it.
There are many societal norms that help us get along with each other better. You do not have to agree with the norm, but it is helpful if you would practice it.
You are making this such a big deal and I'm not even going out with a woman here.
I know you are not dating her, but it is still a nice gesture to give up your seat. People will see your behavior and think of you as a role model.