"Verify language files" just affects static localization, i.e. translation of DNN elements.
It does not affect content localization.

That is good to know so that you can exclude it

Then I have no clue, thought that this point is the problem.
I can't click on the german eye or the gear to remove and add the page, - it redirects directly to the english page

any suggestions?

please check "languages" tab on the page settings dialog for managing localization of module content on this page.

Under the point, at least everything is as it should be. It shows me everything correctly.

Short explanation about the structure of the page,

in the menu we have 3 buttons
Product with some subpages www.xyz.de/en-us/, about us www.xyz.de/en-us/about-us, xyz www.xyz.de/en-us/xyz

The product page is at the same time our start page (www.xyz.de/en-us/), this works without problems, as mentioned above, if I click on the German flag (link shows www.xyz.de/de-de/), we still land on the English page (since this is our main language),

If we go from the menu item to Product - to a subpage let's say Product A (www.xyz.de/en-us/product-a/), and change the language here - we get to the Product A page (www.xyz.de/de-de/product-a/) in German. - this is correct.

Now something strange:

If we change the language parameter from /en-us/ at www.xyz.de/en-us/product-a/ to /de-de/ we are suddenly on the German start page. Even if I edit the link a bit and delete everything from the product-a except for 1 letter, I still get the home page.

Which DNN version are you running?
Did you setup 3 primary Aliases for the website - a neutral one and one for each language?

My Version is v09.10.02 (0),

we got only 2 primary site aliases:

xyz.de/en-us - Laguage en-US
xyz.de/de-de - Language de-DE 

is the neutral one the xyz.de without /xx-xx/ ? - its also Language de-DE
if i set her as primary - the de-de language lose her Primary status.

we have more site aliases, but these are for internal use - like ip address or localhost - but these are not set as primary

there should be a third primary portal alias xyz.de - no language assigned

It works, big thanks for the help

However, I have now discovered a slightly different problem,
It's about the fact that when I click on the flag, from EN to DE it goes with one click (page refreshed once and everything is in German), the other way around so from DE to EN it takes 2-3 clicks.

Should I open a new ticket for this?

I solved the problem - Update to v9.11 fixed it - thanks for the help :)