I don't know a single gamer who rented a domain for private use. In fact, I would argue than the hassle of setting one up is the reason why Hamachi got popular back then. You don't have to bother with knowing any technical stuff, just download a software, share a code and play.

DynDNS was big among some of my gaming friends way back in the 90s at one point, when it was a free, donation-supported tool. It was quite useful and relatively low hassle.

At the time it provided a real simple desktop tool that you would install, sign in to your account name, and it would auto-update a (very) short TTL DNS A record for you. (Generally in the form of username.dyndns.org, but as I recall donators could also bring their own top level domain.)

We've got mDNS today to fill some of that gap, but I still wonder if it would also still be nice to have a "no click" desktop tool in 2023 that could quickly update very short TTL DNS AAAA records for you on a subdomain of your choice, and sort of lament Dyn's many pivots (and eventual Oracle buy out) because that original idea still has legs even if it didn't survive the 90s. (Though maybe this time as a true non-profit internet service or operating system feature.)

People used to use https://www.noip.com/ and similar services for this.

Hamachi was popular back in the day because it meant you didn’t need to forward ports on your router. (Something that still confuses some of my friends).

You dont?

It was extremely common with Teamspeak.