How can I get the "Install app?" prompt for my Discourse

Search your settings for native app install banner and enable them at will.
EDIT see below

Is it possible to trigger this notification so I can see how it appears “in live”?
I enabled the option on my forum and opened it on Android, but I didn’t see anthing.

This is wrong. Native means Android/iOS install banners for DiscourseHub. This is desktop PWA install.

That is enabled by default on every Discourse standard install.

That is not a notification and a website can’t control it. It will appear on the location bar when a compatible site is visited enough times. At the moment only Google Chrome and Microsoft Edge have this feature on the desktop.