So to answer my own question: I think this won’t be possible.
This is how far I got: I ruled out iOS immediately due to lack of flexibility. On Android, it is not possible to isntall all necessary extensions in Chrome or Firefox but there is a Chromium based browser called Kiwi which let’s you install Chrome extensions – that sounded promising! It was possible to install Yomichan and “Language Learning with Netflix” that way. Then you have to install some extension that simulates a different user agent in order to pretend that you are using e.g. Chrome on Windows. Otherwise you would be prompted to just use the Netflix app instead.
HOWEVER, this does not work since Netflix wants some DRM engine and in Chrome, the only available one is Widevine. This is however not supported in Chromium based browsers like Kiwi.
So, tl;dr: I could not get it to work and I don’t think it is possible. The only workaround might be getting a windows surface tablet which runs Windows 10, so it would be treated like a Desktop PC and it hopefully has a proper installation of Google Chrome that supports all the necessary extensions.
(I won’t lie… I’m actually considering getting a Surface for this but I will wait before I make such a stupid purchase just for Netflix on the other hand, I don’t own a TV so I saved a lot of money there and could pretend it is only fair to get a tablet instead… )