"Bourgeois" is a word with a political meaning. It basically means "middle-class." It is slightly derogatory. People do not like being called "bourgeois."
In England, Europe, and the United States, particularly at the time Mann was writing, many members of the middle-class often belonged to conservative Protestant denominations and believed in thrift, respectability, conformity to social norms, and traditional attitudes toward sexuality. For example, someone "bourgeois" would not approve of divorce, even when perfectly legal and amicable.
"Bourgeois" tends to mean "the white-collar class," people who their own homes, people who own small businesses or belong to one of the professions... as opposed to "blue-collar" workers in factories.
"Civilized society is one huge bourgeoisie: no nobleman dares now shock his greengrocer."--George Bernard Shaw
"My grandfather... rose to distinction in his calling... and when he was referred to as ‘a highly respectable bourgeois,’ resented the description. My grandmother remained to the end devout and unambitious, occupied with her Bible, her children, and her house; easily shocked,..."--Robert Louis Stevenson