I'm sorry that it's difficult to discuss these issues reasonably based only on simplified analogies. Again, these are real issues with potentially real police or courts involved if it got serious enough, so there is only so far it's sensible for me to go with any examples.
Yes, the situation absolutely was a practical way of preventing fraud. It saved us a significant amount of money with no apparent downside except for a little time to implement the security measures and the slight GDPR concern we've been discussing. The pattern we were looking for in that case wasn't quite as simple as the name example, but perhaps you'll take my word that it really was almost as obvious but it did also have personal data/identification implications. As I wrote in another comment, it's amazing how dumb people are sometimes but even dumb people can still cause damage. I have a few other examples in mind where similar principles apply and those have also prevented material damage to the business and/or other customers.
Just to explain one detail that might look implausible, the grace period being exploited wasn't for new customers, who do have to pay up front. It was for existing customers who pay late (or, as it turns out, sometimes not at all) when further payments are due. Ironically part of the reason we allow that period beyond wanting excellent customer service is for GDPR compliance. We have an obligation to protect any personal data we hold properly and there is at least a plausible argument that deleting everything the moment an account goes overdue on a bill would not meet the standard.
As you have perhaps guessed, this is a smaller business and we do try to be a "nicer" one. Most of the time I think that is a good thing. However it does mean we don't have dedicated staff or budget for any issues like this. When someone on the far side of the world is trying to rip us off, one of us doesn't get to sleep that night until we've fixed the problem, if we can. Every time we have to spend time and money on compliance changes or taking professional advice and every time the business loses money to fraud, that has rather direct consequences for the personal finances of the people who are doing the work to run the business. We do take security and privacy seriously and we try extremely hard to stay on the right side of any relevant rules (far more than most professionals we talk to expect for a business of our size, and I'm told far more than a lot of much larger businesses with dedicated staff for this stuff).
But it really does boil my blood when people say things like GDPR compliance is easy unless you're doing dodgy things or they assume that because I don't agree it means we haven't thought about it or run a business professionally. If the issues were so simple and obvious, there wouldn't be 16 comments under my original one as I write this without a single citation of either the GDPR or any regulatory or court authority to back up any of the answers given or claims made.