The other comment is not quite right, a ‘language register’ is not a dialect, even though apparently the whole classification is difficult and imprecise due to the nature of languages as a continuum. A ‘language register’ means a variant, choice of words, that are used in specific situations or settings: https://en.wikipedia.org/wiki/Register_(sociolinguistics)

So, afaiu an ‘informal register’ would be something like brospeak or language spoken at home and among friends, contrasted to that spoken with strangers and at work. But I don't know what the situation is in German. With English and Russian, every generation and each subculture invents their own slang just to differentiate themselves―can't imagine how any country would avoid developing informal language, considering the existence of Oktoberfest.

> what this refers to

I'd assume it's about the dialects, which are very different from the "official" language, so much that Jerome K. Jerome had a joke in one of his books, more than century ago:

"Germany being separated so many centuries into a dozen principalities, is unfortunate in possessing a variety of dialects. Germans from Posen wishful to converse with men of Wurtemburg, have to talk as often as not in French or English; and young ladies who have received an expensive education in Westphalia surprise and disappoint their parents by being unable to understand a word said to them in Mechlenberg."

"Modern times" and technologies suppress the dialects and with each generation the portion of the local-specific dialects is being lost.

In German, speaking in different registers actually changes the verb grammatically. Kanst du vs Können Sie vs Könnten Sie.

Could you (low register to someone of equal or lower status) vs Could you (medium register to someone of higher status) vs Might you (highest register to someone of higher status). In English I had to change to a different verb, non grammatically. In German these are all the same verb with a different rule applied.