So let's say I launch TrackMyEatingHabits.com. If a user that happens to live in the UK signs up, they can send me a request for all of their data and I have to comply, right? So this means that when I design the software for TrackMyEatingHabits.com, I should also be mindful to have a process (and data model) that makes it easy to locate this user data quickly, right? Also, I should have in place processes to verify…
To put it another way, are you worried if your US company breaks Chinese censorship laws? That's illegal in China. Are you worried if your company denies that the Holocaust happened? That's illegal in Germany.
You only need to abide by laws in the country you are in. Unless you have a EU server you don't have to abide by EU laws.