Opening the car door for the woman on a date is a purely secular custom. A Yirei Shmayim doesn’t engage in that goyish copycat practice.
It is similar to the secular culture custom to open a building door for a woman and allow her in first. (Which is even worse, since a man cannot walk in behind a woman.)
In fact, by Yidden the correct practice is for a woman to allow a man to go first.
If someone here thinks opening a door for a woman is a matter of ‘respect’, please explain why the standard practice should not be for the women to open the door for the men as a matter of respect.