In the statement "Hollywood was looking more and more like a Toys 'R' Us outlet", is there a name/term given to the "more and more"?
The words more and more in this phrase are an adverbial phrase. Being adverbial, they describe the verb, which is to say, they describe the way in which Hollywood was looking like a Toys 'R' Us outlet.
Oxford Dictionary gives the phrase more and more as synonymous with the adverb increasingly:
To an increasing extent; more and more.