Translation:January is usually the coldest month.
"Most cold" is certainly proper English.
In the superlative, almost all (if not all) adjectives can take the construction "most + adjective." So, the most cold, the most delicious, the most ridiculous. In order to take the construction "adj + -est," the adjective itself may have to be changed, and some adjectives will not work. Of the three examples I gave, only "most cold" also takes the form "coldest." Between the two options, some usages are more common than others. You would say and hear "the coldest month" more often than "the most cold month," but it is still proper English.
I don't know if you're a native English speaker, but no-one would say "the most cold month." It's not idiomatic, though it may be grammatically correct in a technical sense. It's the kind of thing a foreign speaker of English would say, who had learned English from a book but wasn't used to speaking it.
That's a very silly ( and dogmatic) thing to say. I was born and educated in England to tertiary standard. I am now an Australian citizen and I'm 72 years old. Your first statement of "no-one would say 'the most cold month'" fails on those grounds alone. I do say that. The idea that I learned English from a book and am not used to speaking it is even more nonsensical. I really do not understand why there are those who feel it behoves them to insist on a version of English that is structured to either their own personal beliefs based on limited experience, or according to some manual which may or may not be helpful to the non-English speaker. The objective is communication on a level that can be widely understood. There will be variations, not just in different countries but even in different states and, in the case of the UK, different counties. Get over it. There is no universal standard for English, nor (in my view) should there be, but please don't try to instruct the native born in what is or is not "idiomatic".
Another irritating question which says you answered incorrectly based upon the most petty insignificant differences between yours and the "correct answer".
I answered: January is usually the most cold month ...
Yet, I said I was incorrect and said the correct answer was: January is usually the coldest month.
My answer should've been accepted. This fussy and picky AI is just irritating...