GDPR protects EU residents (even abroad) when a business sells to them (or when a website targets them). At the point of sale, in order to figure out whether or not GDPR applies, a business needs to figure out whether or not someone is an EU citizen.
You have a couple of choices with a law like this:
1. Just comply with GDPR anyway. That's honestly the easiest choice, especially if you're already privacy conscious. But you're lucking out, because GDPR is a relatively mild law and comes with a bunch of exceptions that make it easy to comply with. It's not a good long-term strategy to say, "I'll just comply with every country, and that way I'll never need to figure out who my customers are."
If you're not interested in complying with GDPR, then you have to stop selling to EU citizens.
2. At the point of sale, you can use something like billing information to try and figure out where your customer lives and block them if they're an EU citizen. This is unacceptable to someone who wants universal anonymity for citizens, because it requires billing information to be tied to identity/location. You're basically guaranteeing that you can't ever move to a payment system that doesn't provide that information.
Maybe you can skip billing information, and use some kind of government ID number instead. But no matter what, you need some way to tie the thing giving you money to the person who legally has a citizenship in a country.
3. If you don't want to verify, you can just ask the person if they're European and block them if they say 'yes.' This is probably the compromise that would make anonymity-advocates happiest, because it doesn't require any extra data to be collected and customers can lie. But that's also the problem -- customers can lie.
In the US, the most direct analogy here is COPPA. COPPA is a set of privacy restrictions for what information can be collected about children under the age of 13. There are traditional ways you can fall foul of COPPA (some sites are just obviously targeting children). But for the most part, the US went with option 3 -- you ask people their age before they sign up for your site, and you block them if they're under 13.
Again, option 3 is great for people who love anonymity. But it takes all the teeth out of COPPA, because children just lie and use the services anyway, and then their privacy gets violated. And the company winks and very coyly says, "Oh, we had no idea 10 year olds were signing up for Facebook. It's not our fault."
If you wanted a COPPA that did more to restrict data collection, you would probably prefer something like option 2 -- where we collect enough information about children so that they can't fake their age, and use that to block access. Except doing that reliably would require either building a national identity database or collecting other data that would itself be considered by some people to be a violation of privacy.