Progressive web app

No, you can't, at least not as useful PWAs.

Let's first take a look at the definition of Progressive Web Apps:

A progressive web app (PWA) is an app that's built using web platform technologies, but that provides a user experience like that of a platform-specific app.

Like a website, a PWA can run on multiple platforms and devices from a single codebase. Like a platform-specific app, it can be installed on the device, can operate while offline and in the background, and can integrate with the device and with other installed apps.

Source

I'll start with the "can operate while offline" specification - that is pretty much impossible to achieve on user's side with many web apps, as they rely on pulling data frequently from the internet. There are PWAs that ignore this and don't even handle lack of connection properly, but I wouldn't necessarily call them fully proper PWAs.

Then we have integration with systems. For example with notifications. This and offline handling is done with help of service workers, which you can't easily define as a user for any sort of websites either. It's also optional in theory, but if you skip such features, there's not much point of PWAs.

And lastly the most important part - ability to install it as an app on various devices. This is done by linking a manifest in the HTML, e.g.:

<link rel="manifest" href="manifest.json">

The manifest defines basic PWA info, such as icons, where the application should start, how it should be displayed, etc.

This part could perhaps be achievable by dynamically adding the tag with your own external manifest through userscript, but once again, I see no point in doing that and it's not even worth trying as adding a website to e.g. your bookmarks is basically a much more optimal equivalent.