Especially being that I'm a paying customer. I think captchas are abusive, not to mention a nuisance.

Yes they are, especially because they're hosted externally. I bet it's because you're pretty new. I was about to say it's because you might be using TOR or somesuch VPN service, as I am, but come to think of it, I've not recently had to do any captcha on GOG anymore, whereas before I had to, even though my account is not new, but was not very active until recently. I do get the 2FA challenge every time though, which is way better IMO. So maybe just enable 2FA and captchas are gone for good? It's called "two-step authentication" in the settings.

Can you not stay logged in?

Captchas were created because of malicious coders making bots and criminals spam logging in these systems. So some genius cybersec coders fought back with this shit. And every bystander user is caught up in this stupid digital war of theirs. It's bullshit.

Huh?

Well, it doesn't matter. I deal with it. Didn't get dinged with a captcha this time around.

Also plausible your ip address changes. As some service providers shift them around. Unless you use any public access points, like an airport. If you are at an airport, see if you can use a vip room. They often have better service.

Because gog needs some sweet sweet google cash to supplement the sweet sweet blackrock cash. Also they (prolly) hate us. Do they have bottle service?

What are the details?

As a new user, I rarely encounter CAPTCHAS, except when redeeming multiple Amazon keys consecutively. I consistently use the same PC and browser, log in a couple of times weekly (Let's say Thursdays and weekend), and always sign out afterward.

You log... out? I logged in and never logged out.

While the wording includes a few personal preferences, it is to the point on a factual basis.
1) hackers try to hijack accounts, for a) plain theft (wallet credits or account resale) b) identity theft (impersonating someone you know, or you yourself, to trick you or someone else or you (resp.) into believing they're someone who can be trusted) or c) post under a reputed name misinformation, advertisements (including propaganda) or just disruptive things to clog the forum. These events usually rise in weeks leading up to elections or on weekends / holidays where the staff isn't on duty.

2) to counter that, Captchas were invented, because 2FA wasn't yet widespread and just isn't feasible for things where you're not supposed to require an account. The harder the Captcha is to use, the better security it provides, but obviously it becomes more bothersome for the legitimate user. Think of it as the key to your front door, paired with someone asking you stuff just to be sure your voice actually matches before removing the bar from the inside.

So ultimately, it is because a few overly selfish individuals are constantly and massively abusing any open system, any open system is eventually forced to spend effort and resources to become a closed system, just to stop the abusers from taking it all away / destroying it for everyone. An inevitable fact of humanity, think of the door key analogy above.

And yes, the stores store your IP so they can check them against a blacklist (or build a graylist): if an unusually high number of logins come from the same IP address or range, and especially if there's an unusually large number of failed login attempts from said IP / range, then everyone using it in the future couple of days / weeks will have to solve increasingly aggressive Captchas get blocked outright if the abuse is too massive still. Also, as you might know from your bank login, if you fail your own login a couple of times, they will block you and have you call them to unlock. Similarly, GOG and others will send you a Captcha to at least make sure that there's a human supervising the logins, or at least increasing the cost of continuing the attack by forcing the attacker to use a Captcha-solver (that's why they become increasingly aggressive if you fail a couple of them) or massivbely slow down the attack speed.

I do get captchas from time to time despite using the exact same IP since my initial registration on GOG, more than 10 years ago (in the meantime I changed ISPs, but kept my IP in the process).

Why would you even want that? Self-hosting without relying on dynamic DNS services? A business contract that you also use for private stuff? But even then, DNS should take care of IP changes if you switch providers (unless you're operating said DNS servers, that is, of course).
I've even set up my router to reestablish the connection once a day solely to ensure that my IP changes in order to hinder trackers.

Good to know that it still can happen. Thanks for sharing.

I do indeed host many services, including my personal mail server. But even without that, I don’t see why I would want a dynamic IP, I see no benefit compared to a static IP.

My reason is trying to retain the last shreds of online privacy that remain today, or at least making tracking less precise and with higher effort, nothing else.
On the technical side, I'm surprised you actually can take an IP with you, as these are assigned to providers. So they implemented a routing exception just for you, it seems. Maybe the running out of IPv4 addresses made large providers rent out subnets to others, much like phone numbers used to be fixed to providers, so there is a system in place that can deal with the large number of exceptions.

In my experience the IP is no longer used as a reliable way to track people. Things like Web browser fingerprinting seem to be the "trendy" spying methods nowadays.

I’m cheating a bit here, and you did not have all the information ;)

I do not use the IPs provided by my commercial ISP (the one I’ve changed several times), but other IPs provided by a local non-profit ISP (this one I never changed). Technically I rely on a WireGuard VPN so I can use the IPs provided by the non-profit ISP while traffic runs through the commercial ISP pipes.