No, the poet does not actually say that the fog is like a cat. However, he uses the cat as a metaphor to describe the fog. The three things that tell us that the fog is like a cat are:

(i) The fog comes slowly and silently like a cat.

(ii) It spreads over the city and stays for some time like a cat sitting on its haunches.

(iii) The fog disappears after some time just as a cat moves away after some time.

NOTE – Metaphor refers to a figure of speech in which a word or phrase is applied to an object or action to which it is not applicable in a practical sense.