I can't seem to reproduce your bug.
- Which web site are you testing with? (If you are testing with a localhost, can you reproduce the bug with a page in https://pwa.rocks/ ?)
- Do you have any custom chrome://flags set? The behavior you are describing matches the behavior when the "Bypass user engagement checks" flag in chrome://flags is set

Xi and I just came across this and we think we have a good theory for why... if a site is webapk-able but you're using the old a2hs (say webapk install fails and we fall back), you'll get prompted to install. I assume this has to do with different logic between the two and when to suppress it (i.e. we assume if if's webapk-able, we check whether it's installed as a webapk and not if a shortcut was added). I didn't think that was the case but we can reproduce it

I could not reproduce the bug with pwa.rocks. 
But it still persists with my site, which is created with Polymer Starter kit. 
Is there a way to submit logs for analyzing the problem? 

Updates for comment 6: we turned on bypass-engagement check, and after turn it off, I don't see the banner prompted.

Oh, thanks Xi. You're right. I think that overrides the logic.

c#6: both WebAPKs and the legacy PWA flow set the same installed bit so that you shouldn't get a banner in either case.

A variant of this bug is filed once every couple of months and it's almost always due to the bypass engagement flag.

nicolai.immanuel.schmid@gmail.com:
1) Can you reproduce the bug with: https://shop.polymer-project.org/
2) Does the PWA shortcut show up in the app list or just in the homescreen? (I am asking because we are experimenting with a new add-to-homescreen flow)
Just to make sure that I understand the circumstances of the bug you are seeing:
3) When you open the home screen shortcut for your site, does it launch in full screen? (The Chrome toolbar would be hidden and a splash screen is shown)
4) Does the bug reproduce for your site if you remove remove the shortcut for your site and add it again?

1) no I can't. 
2) it is on the homescreen and when I remove it says uninstall, where as an other PWA says just remove. 
3) yes it gets launched fullscreen, with a splashscreen and everything 
4) yes the bug still persists 

Maybe it's related to this issue 

https://bugs.chromium.org/p/chromium/issues/detail?id=735641

I had the same behaviour :
1) No banner in the browser for my PWA
2) Manual add to homescreen
3) Lighthouse told me after an audit that 'start_url' does not refer to an url cached by service worker. Indeed, start_url was '/' and I cached '/index.html' (but my PWA worked when opened from homescreen)
4) I fixed the start_url
5) Banner appears when I open my already added to homescreen PWA 

c#13: that would explain it. Changing the start_url will mess up our housekeeping logic for whether or not you've added a site to homescreen.

WebAPKs should address this in the very near future, allowing the start_url to be updated and the check to work correctly.