Why would they think "its a web-site"? PWAs just look like normal phone apps.

In fact, many "normal" phone apps are basically just a web-site inside a thin wrapper. So the difference is largely academic in many cases.

Because a user first encounters it on the web - where it acts exactly like a website. The only difference is that the bookmark-button-thingy says "add app to home screen" instead of "add to home screen".

It's not giving a "I'm installing an application" vibe, it is giving "I am creating a shortcut to a website" vibes. Apps are installed via the app store, not as weird quasi-bookmarks in your browser.

Most ordinary people thing that they are just adding a shortcut to the website, whereas these wrappers convince the user they are actually installing an app app.