In German it is called "Bürger", yes. Burgher is some weird English spelling of the original french one, and I don't think it applies in any reasonable way to Thomas Mann. In German it really just means "Citizen".
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”.