***** NOT A TEACHER *****
You have asked a great question.
(1) My teachers taught me to simplify sentences for faster analysis.
(a) Let's drop "in Moscow."
(b) Now we have "What's the weather like?"
(c) But in order to analyze a question, we usually have to put it back into 'regular" order.
(d) So we get "The weather is like what?"
(2) I think that most dictionaries and grammar books nowadays tell us to parse (analyze) "like" as a preposition:
The = adjective. (I know only traditional terms. I think that the "modern" term is "determiner.")
weather = noun.
is = verb.
like = preposition.
what= interrogative pronoun (object of the preposition).
*****
Some older books (and older teachers!) say that "like" in your sentence is really an adjective.
For example: The weather is nice/beautiful/ horrible/like/ fantastic/ gorgeous, etc.
Their (old!) explanation:
The = adjective.
weather = noun.
is= (linking) verb.
like = adjective.
unto = preposition.
what = object of preposition.
As you can see, it is very much easier to just parse "like" as a preposition! (Since I am very old, I will stick with the older explanation.)
