Organize Games on Steam or Add Notes to them

I'll comment specifically on Playnite, but I don't believe there is an extension that does what you're thinking of. The issue from what I see is that Playnite can't send any information to your Steam account - it just retrieves your library information. As far as I know, it's not a two-way communication street.

Now, it sounds possible you could create a script that reads your Playnite library information (I believe it is saved locally on your machine) and checks whether the Steam game page you're entering has a match. I haven't looked into it, but Claude/ChatGPT have been quite helpful lately when I've had to code some basic Powershell scripts - it could probably help you achieve something like that with TamperMonkey or other userscripts.

Not sure if that's what you want, but recently I started using GOG Galaxy's library integration to help me manage my collection across multiple stores (Epic, Steam, Ubisoft, even Amazon Games). It's been great seeing what I own and don't own, and it's helped me not buy a couple games I wanted to because I noticed I already had them on another platform (The Messenger for example, which I have from Amazon Games). You need some workarounds for some integrations to work properly but nothing a smart person like you can't handle. Really recommend it, feel free to DM me if you need any help.

Have a nice day!

I use the feature on Steam to "ignore" with the "already own on another platform" option.

I have a similar usecase, but because theres no extension that does such a thing i just alt-tab and search on playnite... on pc of course. On mobile no way...

https://github.com/Depressurizer/Depressurizer would really fit, it was a great tool. Sadly it hasn't been updated in 2 years and i don't think it functions after this long time due to Steam changes and bugs.

Sadly i don't know any other project like that or working fork of it

I manage my game libraries on gg.deals and also use that site to check for best prices before making any purchase. That way accidental double purchases can't really happen. You can sync your Steam and GOG libraries directly and almost everything else through Playnite, although I prefer to add new games manually. It also has a wishlist and price alerts. And if you're worried about the keyshops, you can opt-out of those.

I don't want to hide the games I already own on other platforms. Only add a mark to them, but I want to see them.

If you aren't using the Ignore feature at all, I think you could use "Ignore (played elsewhere)" and disable hiding ignored games.
That way you'll see the ignored games and they'll have the "tag" so you know you own it elsewhere.

I'm using the most medieval way, which is to manually add a custom tag to indicate this.

I'm not sure how default?official? tags work vs custom tags, so I name such tags in my own unique convention so as not to accidentally mess up official tags. Thus, I can see them on the web browser as well. But my use case for this is quite rare as I use desktop/Playnite usually.

Slightly digressing but I really wish one day Steam can let us filter our custom tags as well. There is one page (forgot which) where it lets you click the custom tags as if to filter, but it doesn't actually do anything.