I may be wrong as I am not familiar with BANano and it's counterparts. I want to say about PWA in general.
What the "install" really means based on my experience is a basically a "shortcut".
Unless you have done something "differently" than others, the PWA apps install on a mobile or desktop, it is just loading the "cache".
If you go offline and refresh the "page", sometime it works (depend on how the developer design the app), provided the resources such as images, CSS and javascript are "fully" downloaded.
If later the user closes the "installed app", the cache may be cleared. The user cannot view the "pages" anymore. If a good developer, he may put a friendly page showing "you are offline" or "something has gone wrong".
Take Postman as an example. This is how SaaS company make money and take control of the usage.
The "install" is just a convenient way for a user to quickly access the app (or basically the website) without typing out the url in "online" or internet connected.
I think it is still depend on a web browser engine and the OS behaviour.
On windows, it seems opening the page on a simplified frame without the clutter of a normal web browser.
On mobile, it seems it also have something similar.
Again, these are based on my experience on Windows and Android with Google Chrome as default web browser.
Good things that I can see are data such as login session is stored with user permission and I see notification is also supported.
I never have experience with notification with PWA so I can't say how good it is working.
For me it doesn't provide a good user experience (at least for me) and I believe this is why Apple dislikes it on iOS and MacOS/Safari.
My conclusion is, I don't mean PWA is bad but it has to work as connected.
In some way, it may save user some mobile data as not all the content need to be refreshed when online.