Is Safari, in incognito mode, saving its settings? I have set Safari on board my iPad Mini (first edition) to always run in incognito mode. Every time I start it and go to YouTube or any other video service, I can clearly see, that volume is set to half. I don't know how many times I turned it down to 0, it always is "reset" once I re-start Safari.

This happens only with Safari. Other browsers (Chrome) and all other applications always starts with the volume, it was set, when leaving it. This brings me to a bit mad assumption, that in incognito mode Safari does not save anything, neither browsing data (cookies, history) nor its own settings.

Can that be true?
That is the point of Private Browsing (Safari's calls it Private Browsing, Chrome calls it Incognito). When you use Private Browsing, Safari shouldn't be saving anything at all. The whole idea is that it leaves no trace.