Why is it called 'the month of January' and not 'the month January'? As I was learning German, I noticed they used the latter (der Monat Januar). Why the discrepancy?
Both are correct:
The month, January, is the first month of the year.
The month of January is the first month of the year.
In the first case you do need the commas to indicate a parenthetical explanation. Without them you have two nouns together, and nouns don't qualify nouns. An adjective would work, but neither month nor January have a corresponding adjective.
In the second case we have a special case of genitive called the genitive of apposition. It has the same meaning as the first. The second noun restricts and clarifies the other. A wordy explanation would be to replace the "of" with "that is to say", viz:
The month, that is to say January, is the first month of the year.
It isn't particularly common in English to use the genitive this way, but that is the particular syntactic structure being used here.
I'm no authority on this, but the etymology for the month names may provide a hint:
January etymology probably comes from "Janus" -- a Roman deity; so if it meant "The month of Janus" that may make more sense. 1
February also had a similar etymology; "Month of purification"2;
It breaks down once we get to the months named after numbers (sept/oct/nov/dec), though.
I've often wondered why we named our last few months incorrectly (october isn't the eight month), but that's a separate issue, isn't it :)
IMO, this is grammatically incorrect and a case of periphrasis gone wrong. But it has nevertheless become acceptable through regular use. It should be the month, January and the season, winter rather than the month of January or the season of winter just as it is the year 2013 rather than the year of 2013.
The use of the preposition of alongside a noun usually indicates possession. Phrases such as the PM of Djibouti or the admiral of the fleet can be restated as Djibouti's PM or the fleet's admiral respectively. This doesn't really work with the month of January as January's month does not compute.
Moreover, the admiral of the fleet sounds far more prestigious than the less flowery fleet's admiral. I suspect that the use of the verbose month of January came about for similar reasons. Even the presence of the word month in the phrase is often superfluous.