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thats quite remarkable. Thank you very very much. I need to dig deeper into the function of the plugin "system - page cache" and pros and cons.
For all folks:
The solution was to disable the plugin "system - page cache". The e-privacy plugin now works as expected, the Domain Cookie is set to ".yourdomain.tld" (without leading www).
By the way, it seems that the site response is a bit quicker now, but I didn´t perform speed tests.
Michael - you really did a good job here and quite fast. And as I can see you are not really affected by the EU GDPR, as you are in the US. Thank you and waving to Wills Point...