Greetings,

I bought this game via the website and i see it present in my library from there. However when i open the GOG application on my laptop and go to "Owned Games" tab, I see all my other games but Silent Hill 4. What can I do? I also see the game in my order history. I've paid for it just to be unable to play??

Jast do it a proper way instead of using Galaxy.
Navigate to your library [https://www.gog.com/en/account ] via browser and download an offline installer. You must be logged on via browser as well.