Where do the search suggestion come from?
The (german) help page about search suggestion does not tell me from where I get those suggestions. Is it a Mozilla service? Google? The default search engine (from the settings)?
Actually, I would expect or prefer that if choose a search engine during typing, that I get suggestions from this site. But I guess, this is not the case. So which engine gets used?
Keazen oplossing
Your screenshot shows the favicon for Stack Overflow in two places and is says "Suchen mit Stack Overflow"; that is how I came up with my answer.
The "currently chosen" is the Default!
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In that screenshot the suggestions are from Stack Overflow which was selected when you began typing in the Search Bar. I suppose that Stack Overflow is set as the Default Search Engine on the internal about:preferences#search page, so if you want a different Default switch to a different search engine as the Default.
Should it be Stack Overflow? I have installed the Ecosia Search Add-On, so it might override that behaviour. Currently, I don't believe that it is Stack Overflow, since I typed things in german and Stack Overflow will unlikely contain those texts, as it is an englisch localized site.
As said before, I would make this clear in the documentation and possibly in the Firefox settings.
Okay. I've found the following https://support.mozilla.org/en-US/kb/search-suggestions-firefox :
Enabling search suggestions will send the keywords you type in a search field to the default search engine - unless you appear to be typing a URL or hostname.
So it should be the default search engine. Not the currently chosen.
Keazen oplossing
Your screenshot shows the favicon for Stack Overflow in two places and is says "Suchen mit Stack Overflow"; that is how I came up with my answer.
The "currently chosen" is the Default!