While I agree that usually "class structure" is a sociological construct to suggest divisions within a society--divisions determined by economic, political, and/or cultural traits--there is also the use of "class" to mean high social rank or style. "She's a classy lady." "First class tickets" In this sense, LotR is loaded with "class" attributes. As a mythology of ancient times, it follows the pattern of focussing upon the aristocracy--
Patrician as both
Morthoron and I said earlier--so that we lack much evidence about the huddled masses of Gondor, Rohan, the dwarven kingdoms. (Actually, I have a sense that dwarven culture is far less class-ridden--as in having fewer social divides--than Gondorian or hobbit culture.)
Both meanings can be argued in the story.