Kinda. It’s a thing for any website designed to the Progressive Web App specification. The underlying concept of PWA seems to be that you can build an application on the web that has a lot of the functions of a native application, and that this app can be easily ported to a mobile device. I would suspect that a lot of newer apps on people’s phones are actually PWAs.
So it’s not unique to Discourse. The other sites I’ve encountered that offer it are Gmail and YouTube, but apparently sites like Twitter, the Washington Post, Starbucks, Uber, Pintrest, and Spotify all have PWAs, which can be installed in this way.
Since most other forum software was written before PWAs became a thing, it would not surprise me if this is not available on other forums. They’d have to specifically modify their code and include a manifest to make it work that way.
As for why you’re seeing the icon now? I’m not entirely sure. I didn’t see these icons until fairly recently. But apparently Chrome added the feature in version 76, which came out more than a year ago.
This is the best I know. I guess I’ll ping @codinghorror to see if he knows if PWA support was added to Discourse recently or was part of the code the entire time.