PWAs can show notifications but Apple refuses to implement notifications for PWAs on their platform. You can use Chrome on macOS to get notifications working there but on iOS you're stuck with Apple's decisions. It wouldn't make sense to add notifications for them since PWAs would obviously cut into their profit margin so don't expect PWAs to work on iPhones or Safari in general.

I use my Home Assistant as a PWA and the notifications work well on all platforms I use. Firefox is behind on desktop though, they don't seem to see PWA support as a priority like Chrome does. On mobile it works fine.

Twitter is already a PWA (though they don't advertise it much) which you can install on any modern browser onto your phone. I've installed it in Firefox on my phone and I've never considered downloading their app since. Outlook.com also has some pretty good PWA support. Those are the ones I know off (aside from a few local websites/web apps only available in my language).

I don't know about other social media though, twitter is the closest thing to social media I follow.