Since this is more directed to the gog blues, I'm not sure how to do this. Should I have (or even should I now) contacted support instead?

Anyhow, I was the only gogger in my family (that I know of). So on this PC, although it is a "family PC", all the gog info on it was mine, sign-ins, accounts, wishlist, etc.
My sister used to enjoy the old Sierra games when we were kids, and she saw me playing one of them and seemed interested, so I figured I'd help her out. I logged out of my own account, set up an account for her and then began browsing the games with her tagging what she might be interested in as wishlisted.
However, it turns out that for games I own (even though I'm not signed in), they can't be wishlisted by her on this PC. I'm assuming it is a cookie issue, and the wishlist is somehow saved locally as cookies (not sure why, that sounds like a ridiculous way to do things..might have even been okay if account info was also saved in the cookie). I could probably fix all this up by clearing my gog cookies, but even if that is true, it is still an annoying workaround.

Now I know that gog is cool, and would probably have no problem with an account being shared by family on one PC, but the thing is, I'm often not home for months at a time, and sometimes my sister isn't staying here either, so we couldn't really share the account properly -unless gog is also cool with family in geographically separate locations also sharing an account...but even then, it'd be a bit weird because of payment methods and 2 people posting as one on the forums (if she ever posts). I understand gog galaxy might be a solution for this, but I don't really use gog galaxy so I don't know much about it, and my sister would also prefer to be spared the complications and hassles of using a different client (she's not a regular gamer, and has no knowledge of steam or galaxy or such things).

Also, not really relevant to her (since I don't think she'll be using the forums much), but I've also noticed that replies to me (the little star circle with the arrow inside it) show up even when I'm logged out and someone else/no one else is logged in.

The browser cookies are typically stored in a user's profile folder (I am assuming that you are using Windows). Why not create a new User Account for your sister so that she has her own desktop, browser cookies, etc.?

Try incognito/private browsing.
You may also try different browser.
Sometimes during slow internet these kind of hiccups happen.

The problem is that she's been using the current account for ages now, so I'd have to set up loads of shortcuts and firefox stuff. Incognito browsing worked! It didn't pick up on my owned games, and I was able to add stuff. Thanks for the suggestion!

It is, of course, a temporary solution (because again, incognito browsing would be a bit of a hassle for my sister to do on her own after I leave, so when I leave I'd probably have to clear my cookies for gog, and then maybe clear hers or something more complicated when I come back), but for now, what I wanted to do is done!

You can place a desktop shortcut for gog.com on your favourite browser's private tab.
Only problem would be loging in everytime.

Not an issue now, but for the future, how would I do that?

I'm currently using firefox, and it doesn't actually have an option to open a private tab, only a private window. When I open a private window, if I create a shortcut to gog.com and place it on the desktop, it doesn't open it in the private window.

Add the -Private flag to the shortcut.

I cant explain it from mobile but here is url that arm help you wit private shortcut http://www.7tutorials.com/create-shortcuts-start-any-browser-private-or-incognito-browsing
https://support.mozilla.org/en-US/kb/create-desktop-shortcut-website

Firefox allows you to have multiple profiles, and you can use this feature to make a custom Firefox shortcut that will launch an instance of Firefox who's history, bookmarks, logins and cookies, extensions, themes, saved passwords, etc. are completely separate from those of Firefox when launched the regular shortcut.

Just take the Firefox shortcut that you normally use to launch your browser and make a copy of it. Put the copy wherever you like and rename it however you see fit. Then go into the properties for that shortcut and look for the "Target" property in the "Shortcut" tab. Decide what you want to call your new profile (the name itself isn't that important. the regular profile is named "default"; for the purpose of demonstration I'll assume you want to call it "babark"). Add the following argument to the end of the "Target" property:
-p babark

Close any open instances of Firefox and run your custom shortcut. Because the profile specified does not exist the Firefox profile manager will appear. Use this to create a new profile with the same name as you used in the shortcut (babark, in my example above) and remember that it's case-sensitive. Ensure that the "make this the default profile" checkbox is unchecked and hit the button to proceed. You should be rewarded with a fresh Firefox with no history or extensions or anything. It's your own personal profile which you can use and customise however you like without affecting anyone who launches Firefox through the regular shortcut.

Just make a copy of firefox shortcut and rename it to GOG (or anything)
Right click and open properties and in target box after "C:\Program Files\Mozilla Firefox\firefox.exe" add -private http://www.gog.com

It should do the trick.

You can also make a separate profile as Barefoot_Monkey said

in that case it would be "C:\Program Files\Mozilla Firefox\firefox.exe"-p babark -private http://www.gog.com

It's a cookie problem that gets worse when the connections are slow. I don't know what the problem is, but the cookies just won't register. I'll have to see if a private tab helps, but it's gotten really ridiculous.

I generally have to login, close the tab, open a new tab and re-login in order to get in.

It's clearly a cookie thing and I'm not even sure when they're going to get around to fixing it. They seem to be far more interested in that pile of crap that is Galaxy.

Brilliant! Thanks for that, Barefoot! And amrit!

I think I might do the separate profile thing. It would even help with me staying signed into gog and gmail and all the other sites, while the rest of the family can stay signed in as well.

Creating a new profile works great! A pity you can't have 2 instances of the browser open with different profiles simultaneously. Would have been really convenient if like someone wanted to use the PC, but I had some stuff open and didn't want to get rid of it, I'd leave it there, but they'd use the shortcut leading to the other profile

You could install Pale Moon, import the Firefox profile so that all the bookmarks and whatnot are present, and then run one (or both...) of the browsers in private mode. Since Pale Moon doesn't use Firefox' profile directories, but (optionally) imports the (default?) Firefox profile into its own, you can have both running at the same time with no problem.

Maighstir's solution is a good one.