The main purpose would be to simply be poetic, and use an interesting metaphor for the fog. Cats are known to be quiet, stealthy, adept, and able to scale and inhabit places that other creatures can't. So, comparing the fog to a cat is an apt description because fog too is silent, stealthy, and creeps into places that others can't. Also, in this beginning part of the poem, Eliot is describing a run-down part of the city; he mentions the "half-deserted streets", the "one-night cheap hotels," the "sawdust restaurants with oyster shells." He emphasizes the run-down atmosphere by using the word "yellow" repeatedly; this is no beautiful fog that creeps beautifully before a glowing moon. Describing it as yellow emphasizes the industrial, polluted nature of the fog. Most likely, in the city, the pollution from industrialization is so dense that it turns the fog yellow. That this yellow, dirty color brushes against the windowpanes indicates that it is right at their door, and hard to escape. Eliot was a modernist writer, and modernists tended to bemoan the dehumanization and pollution of the industrial age, so Eliot's mentioning of the "yellow" fog plays into that.
Also, the part of the city that he is describing is rather decrepit, and with that image comes the fact that there are probably many stray cats that are running around in the alleys. So, as the fog creeps through this part of the city, alley-cats are probably fairly common and a part of that run-down atmosphere. So, rather than comparing the fog to say, a snake, Eliot picks an animal that is regional and logical for that part of town.
I hope that those thoughts help a bit; good luck!