It meant an upper middle class urban citizen, while "Kleinbürger" was their lower middle class counterpart. Buddenbrooks was all about Bürgers, their history and lifestyle. Mann was a member of that class or even of its upper crust, the patricians.
This isn’t hard to understand. “Burgher” is a perfectly legitimate translation of “Bürger” as in “bürgerlicher Mittagstisch”, “Der Bürger duldet nichts Unverständliches im Haus”. “Citizen” is a perfectly legitimate translation of “Bürger” when it comes to “Bürgeramt” or “Weltbürger”.
It most definitely does not — it’s both “citoyen” and “bourgeois”.