The "app" is what is called a PWA or Progressive Web App. It's basically an advanced bookmark for lack of a better term, with the ability to process push notifications.
I've never seen that warning/popup before, but I haven't used an Android phone in many years. Any chance there was a recent Android update before you started seeing that popup?
Also, we haven't used that icon for at least 6-7 months, so I'm wondering if you have some stuck caching going on or perhaps stricter than usual security settings that are causing issues. It may be as simple as un-installing and re-installing it.
FWIW, the good news is we're exactly 8 weeks away from Black Friday, so you can start watching for deals to upgrade to an iPhone and solve this and so many other issues! 😈