This is somewhat similar to Bug 1826147 I opened some time ago.

We are experimenting with a new approach to getting user consent to diagnostic data collection (which is required by AMO) that wouldn't modify any existing tab (we used to put a dialogue to the active tab) because of the issues that can cause. We're opening a new tab right after the addon is installed instead, the same way as many other addons.

However, our addon changes the default search engine using chrome_settings_overrides.search_provider and opening the new tab causes the confirmation dialogue for the search engine change to close. There's no way we can open the dialogue again later (or it's not documented). That means we can't both change the default search engine and ask for data collection consent (or display a „Thank you for installing our extension! Here are some tips on how to use it“ page, which is a widely used practice).

IMO, the correct behaviour would be if programatically opening a tab wouldn't close the dialogue.

I've attached a screen recording of the issue.

This is technically a duplicate of Bug 1544271.

The issue is hit because this doorhanger is hooked to the urlbar and so that doorhanger get closed automatically when the current tab is switched to another one.

In Bug 1784292 we were moving it to be anchored to the extensions toolbar button, but we had to revert that in Bug 1789407 to fix a regression (because the add-on post install doorhanger, which is currently anchored to the app menu button, was hiding it).