Is there a way to "Save Favorite Games" on this website?

Yeah, there's a way you can do it.

Go to the Report Analysis of your game >> Then on right side menu, tab 'Analysis' >> Save (here you see also the 'Saved Analysis' button)

With this you can save the games you believe deserve a star (or for whatever other reason)

It hasn't been released yet but is called Library: https://www.chess.com/article/view/chess-com-month-in-review-october-2020

That's not what it is. It's a new feature, and it's ALMOST there. 

Hope it will work on mobile too, not just desktop. 

There is a way to save games. It's called the library. You can create multiple folders (called collections) in the library. 
Open the game you want to save and then click on the library icon and select the collection in which you want to add the game. Look at the attached images.

You can access these saved games later from the homepage. Hover on the 'Play' icon then click on 'Library'. See attached image

You can also save other players games and add participants by sharing the collection. To know about this visit this link. If you just want to save your own game then no need to visit the link.

https://support.chess.com/article/1987-how-do-i-use-the-chess-com-library

I'd like this feature too. Wonder if it has been released yet.

It's described in the post above yours. It's called "library" and was released in 2021. Very handy.

Yay! any way to save your favorite puzzles?

You can save the puzzle PGNs in a Library collection, but they'll just be games at that point.

Any update so far? x

Any news on this subject?

The solution is saving game to a Library collection.

This feature is not available in the mobile app right?

That is too bad, because 9 times out of 10 I am playing in the mobile app. I found that it happens frequently that I think "that would be an interesting game to analyze, but I don't have time for it right now". But it sounds like it still isn't possible (or no easy way) to save that game for later? (on the mobile app that is).

If this is not easily doable, then this could be a factor for me to consider Lichess. There you can simply "star" a game in the app.

That's not an option on the apps.

Hello, I'm looking for this feature. Is it still in development?

The solution created was the Library feature.

https://support.chess.com/en/articles/8609806-how-do-i-use-the-chess-com-library

Thanks