Are PWAs "only" websites that are shown as an app or do they have features that would distinct them from "normal websites"?

In theory, they're the same. In practice, user agents (read: browsers) can dictate extra, uh, privileges to PWAs. For example, Safari Mobile won't allow push notifications for just websites. As a dev, we don't control this aspect. We just code "isPushAllowed()` and continue from there. As a marketer or apps/software I learn what browsers and environments allow what, so I know they need to install as a PWA to get access to some APIs.

Can PWAs reserve disk space for offline downloads for example (Spotify has got a PWA but that one is mainly their website shown as an app).

Branching off the other point, it's up to the user agents. For example, Safari Mobile will delete storage for websites after a certain time (I think 6 months). But PWAs have unlimited lifespan (a misconception years ago). It's not what you asked (storage size vs lifespan), but it gives you a clue to how browsers manage this. I personally haven't experienced much issue with storage, but I do know there's even a File API which gives access to your actual drive. I would imagine if browsers are allowing us to access "unlimited" storage over the file system, the restrictions for the Local Storage aren't that important.

A short search brought me here which mentions 80% of total disc space for Chrome, 50% for Firefox, and 1GB for Safari. Nothing here talks about PWAs, but I'd imagine Safari is the only one worth concern. I can research more if you want, but it's not in my use case to concern myself with local storage.

Edit:

And I probably can't install a listener via PWA like Discord does for their Rich Presence, can I?

As long as there's an API, for better or for worse, we can do it in PWA. From what I read, there doesn't seem like anything like this in the Web API. Background tasks are currently somewhat restricted in the interest of battery conservation of mobile devices. From a theoretically point, an app running in the same environment as the PWA could have somethig shared between them. There are API in addition to Web APIs, like extensions, but there's probably a lack of normalization for detecting what is running on the device.

The idea of making everything in the Chrome Addon/Extension API in to real API is part of Product Fugu and from looking Chrome never built anything to expose what native system processes are being run with a web site. It's honestly such a specific (and invasive) prospect I doubt it'll ever happens. The closest is chrome.tabs but there's no web equivalent.

Sharing of other PWAs running probably won't happen because sandbox/isolation is pretty core to Web right now. Cross-app data sharing is something I don't know about yet, but maybe if PWAs mature we'll see that. Then games and media players can "broadcast" to other PWAs data like presence. I'd imagine that branch from the Web Share API.

I'll take this opportunity to state my bias. I believe in the Web standards, meaning it shouldn't matter what browser or environment you use. They should all be the same. I want this for myself and express this to my clients.

But because standards are "living" it's always a moving target. What is supported today by your preferred setup may not take full advantage of what a Web App provides. But the idea is, after a while, you will gain those features later in your setup as browsers get better.

This is something I've seen in practice since I wrote a lot in PWA and told my clients that eventually it'll be in Safari, but I'm not going to write a native app for outdated browsers. And the investment paid off. Before, Safari users didn't have Wakelock, Push, or good WebRTC experience, but today they're mostly indistinguishable from Chrome Android. I didn't have to change a single line of code to make that happen.