The "of" in "the month of January"

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.