First, thanks a lot guys for your quick answers.
I tried to find help on several SEO forums.. But yeah... the most of them are just without any activity.
is your hostname on a gTLD or a ccTLD
it's a gTLD one
would also look at everything in Google Search Central's Managing multi-regional and multilingual sites [developers.google.com] documentation
Already did it before asking for help. We have integrated years ago /lang/ in the URL and are using hreflang tags
i would start looking for clues in the International Targeting report [support.google.com] in GSC.
I found the link to access to this tool on the "Legacy tools and reports" on GSC, but have this message:
"Legacy reports and tools are not available for domain properties."
Is that just on my side ? or did somebody have this issue also ?
What is your definition of a German visitor?
Well, when I'm checking SEMrush KWs positions on German Market, I can see that some branded KWs are here, but the linked URL is the English one, and not the German one.
The alternative language pages must be indexed before they can show up in the SERPs. Did Googlebot properly fetch and process these pages?
Yes, all the translated pages are indexed. I checked it on GSC. I can even find these pages on the SERP when I'm trying with some alternative branded KWs.
The weird thing is that, since some days, google looks like started again to show the right pages on the SERP... but looks like he can't keep them in the right way lifetime. And GA confirme these swicthes. When a branded KW is linked to the English page again, I can se a drop of my alternative language page traffic and the English one is incrasing.