I see 2 issues on your website:
First, page 125 is in German, page 141 is in English, but there are not translations of each other. Thus you can't benefit of all features.
Second, you use menu-deutsch as menu for both German and English. You must use one menu per language. If your menu contains a link to a German page (here Produkte), even if you are browsing an English page, you will be directed to the German page. That's why you need to create one menu per language with the correct pages links.
And for your information, Polylang generally does not rely on cookie information except some rare cases (for example to set the language on login page).