When I try to wishlist on the GOG games portal, I am asked to log in again and a message comes up with a bear holding up a planet saying that the system is overwhelmened by the popularity of GOG's service.

Ah, the GoGbear is back!
Well, don`t panic, sooner or later it will work again.

try https://support.gog.com/hc/en-us/articles/360021998113

WELL I'LL BE DARNED! It worked. Thanks mate. All the best.

I have the same problem. I'll try the workaround.

This issue appears to persist 5 years after the original post, though the linked support article has been removed and the workaround does not appear effective at this time. Games can be removed from wishlists but not added to them.

If it's cropped up for you, it was either a temporary instance on GOG's end, or it's something to do with your browser (or your connection to GOG). All methods of wishlisting (clicking the heart by a game on the front page; hovering over a game on the catalogue page and clicking the heart on the pop-up panel; clicking "wishlist" on the product page; and moving an item from cart to wishlist) work fine for me on Firefox 115.9.1esr, and I don't recall having had any persistent problems with this in years, if ever. Maybe check in a different browser?

No such problems on my side (Win 10, Edge).
Not now, and also not five years ago.
I can add and remove games from my wishlist just fine.

Out of curiosity: how many games do you have on your wishlist?

Maybe there exists an (invisible) threshold, of how many games can be wishlisted in total, and you crossed that?
(though: I've seen people with literally hundreds of titles on their wishlists...so, it's probably not that)

No problems here either, Win 10 Firefox.
But I do know if you have issues with assets loading on the site, whether it's a CORS error, timeouts or blocked elements and the backend scripting (or login creds) doesn't load correctly, then you get those situations where clicking the wishlist save button fails and asks you to log in. The assets needed for the wishlist are often the last loaded for a game page... individual results may vary I guess, but I've found if I'm too quick hitting add to wishlist while the game page is still loading, then it fails.

So it's still likely a problem on your end, Snakkin. Sorry, but it does work fine for me and others.

It would be nice if it's a local user problem, either cached pages or cookie certificates, or some minor script incompatibility with a browser.

But i suspect there's been some corruption on the Database as little things keep cropping up here and there it seems. Nothing too serious as i'd imagine payment and ownership is handled by a different system than wishlists and todo pages; But enough to cause some problems.

I'm having problems with this currently as well, using 3 different browsers all doing the same thing. I click Wishlist and it turns that orange color for a split second then reverts back to the black "Wishlist it" and if you click it a bunch of times it will eventually save but nothing is added to my actual wishlist. I am at 75 games so maybe thats a limit?

Edit: Just tried to remove some games from my wishlist and apparently thats broken as well for me. Not sure what I can do here.

Still no problems in that regard on my side.
Can add and remove games from my wishlist just fine.

Nope.
To not make some outdated statement, I tried crossing that number successfully just now, and still could add and remove games without fail.

Apart from that: there are users here on GOG with several hundreds of games on their WLs.