I didn’t before to but started using mouthwash after black coffee as I don’t want acidity to destroy my enamel
I usually brush my teeth after a couple of cups, but not necessarily after all the cups.
If you’re worried about the enamel on your teeth, you should switch to an enamel repair toothpaste. Sensodyne Pronamel Enamel Repair toothpaste is exactly that. It’s 7.50 a tube, but it’s’ really good toothpaste.