A lot of people would be happy not having others reminding them all the time about their mental condition

Don't let GAD define your life. Speaking about it all the time and blaming everything on it will convince your subconscious mind that failure is the only option due to GAD . It's not. 
You have the control not your anxious mind which is not you

I don't see anything wrong at your mother's behaviour

For weeks I've been telling my mom about how nervous I am when I'm driving. She told me, that there was no reason to be nervous and "What's the worst thing that can happen?".

Well today I failed my drivers license exam because of my anxiety. I told my mom the reason why, and she asked why I was nervous. I told her "I have a mental illness, that makes me nervous all the time."

And y'know what? She straight up forgot, that I have been diagnosed with GAD. She was even there, when my psychologist told me. Ugh!

Failed my drivers license & mom forgot about my anxiety