Sounds like an instance of what is called the adverbial genitive, of which even English has some vestiges. I believe "I work nights" and even the word "sometimes" (note the "s" at the end of each, English didn't always have the rule of the apostrophe to mark a genitive "s") are instances of this.
This is interesting, thank you. I have never heard of this but it makes sense and fits in with German and the roots of English. I had always assumed that "I work nights" and "sometimes" were a pluralisation (and in both these examples a regular occurrence implies the plural). Perhaps the key is to find a phrase with an appropriate noun with an irregular plural. I must look into it further.
Aren't those examples simply plurals... rather than possessive / 'genitive'. Because "I work night's" doesn't make any sense to me. It's like, I work each night -> all the nights -> the nights -> nights. But you sound like you have some deeper knowledge! Would be interested if you could provide a grammar page or explanation for how these are examples of (archaic?) genitive case.
This leads to to the advanced topic of demonstrative pronouns (yep a simple der has such complicated grammar) ....but a simpler answer for now is that "I like the Friday" doesn't make much sense unless you know which Friday we're talking about eg"Thursday the 2nd is awkward but I like the Friday" in which case "that Friday" is a bit better english anyway
I like the Friday and I like that Friday are different. In the first clause "Friday" is emphasized. In the second clause "that" is emphasized
eg. I like the Friday rather than the Thursday I like that Friday which happens to be the 20th June rather than the Friday which happens to be the 27th June I assume that this distinction is not made in German.