These are the things you need check to determine whether your progressive web app is installable, and not just a web app shortcut to a user's homescreen.
After adding the service worker and a web manifest to a web app, many people think that now, oh yeah, we got a progressive web app. Well, in most cases, that's not.
When a visitor visits your web app for the first time, they should be automatically promted to install your PWA on their devices, that is; destop or mobile.
One quick way is go to your apps menu in your device's settings. It should be listed as an installed app. With this, man, you are at a safe place.
The second one is to try and uninstall from your device. If it just disappers, am sorry. That's a "shortcut progressive web app." The native uninstall dialog should popup like for any native app installed on your device.
Lastly, while your progressive web app is installed on your device, switch off internet connection, check whether it's working offline. To confirm that, restart your device and launch your progressive web app. You should be able to view the app.
To get you started, checkout a detailed progressive web app checklist available on this dev site doc by Google Developers.
Finally, use Google's progressive web app audit tool. It has audits for performance, accessibility, progressive web apps and much more. Get started here.