'class'does not usually mean wealthy in the United States.

In U. S. slang, a person who has 'class' is 'stand-out,' 'first-rate,', 'top notch,' distinguished, 'elegant,' has superior qualities that put them above the crowd.
"She's a class-act," "she is classy."
"She is classy" means she knows how to dress with style, has good taste, is a 'beaut,' etc.

It also means a subdivision or group, e. g., "he belongs to the class of 1960" (the year he graduated from school, etc.), "he is a Brahmin," e. g. a member of one of the old dominant families in politics, money, education, usu. 'back east' (the Cabots Lodges, etc.), "he/she belongs to the upper class," e.g., well- educated, or has wealth, or is well-mannered, reads literature (Milton perhaps but not "True Detective" or comic books and subscribes to the NY Times but not a tabloid), etc.
"He belongs to the lower classes," e. g., he is 'blue-collar,' a pool-hall lout, has an immigrant's characteristics (obsolete), has no importance, no education, etc., etc.
"He ain't got no class" (applied regardless of income status), e. g., lacks finesse, doesn't know how to behave in 'polite' society, etc.

Many shades of meaning depending on the speaker or situation.