This has happened to me before, but this time I can't seem to get around it. I ordered a game from GOG, but it's not showing up in Galaxy. It shows on my recent purchase list, but not in my owned games. Please help.

Have you confirmed if the game's in your Library (using the browser, not the Galaxy client)?

You can try to refresh.
https://www.gog.com/account/refresh

9/10 someone posts this they bought it on a different account for whatever reason, but it's happened to me before.
If the refresh doesn't work then you're stuck contacting gog, with a ticket or messaging gog staff directly.

Also, if it's a goodie package it typically won't be in galaxy.

I ended up fixing it by doing a manual add, but I am not sure why I should have to do it this way. And yes, the game showed online in the GOG website, just not in Galaxy.. Next time I might try a refresh.

But you DON'T have to do it that way. You don't need the galaxy client to access your games.
Just use the normal installers in your account here - https://www.gog.com/en/account

Are you 100% certain you're logged into the same account in Galaxy? The last time I read someone having this problem that's what the real issue turned out to be.