In Around the World in Eighty Days by Jules Verne, why is it doubted that Phileas Fogg is a Londoner?

Because he appeared to have no financial, social, and political interests in the city, it was doubted that Phileas Fogg was a Londoner.
For example, Phileas was never seen at the Royal Exchange or at the Bank of London. He appeared to be unemployed (he was not a "manufacturer, nor was he a merchant or a gentleman farmer"), had no ships in his name at the London docks, and was not a member of any of the Inns of Court in the city. Inns of Court were professional associations for barristers in London, and Phileas was never seen at any of them, whether it be the Temple, Lincoln's Inn, or Gray's Inn.
Additionally, since Phileas never graced the courts or government entities in London with his presence, one can assume that he was never involved in the political and fiscal aspects of the city. Accordingly, Phileas' voice never "resounded in the Court of Chancery, or in the Exchequer, or the Queen's Bench, or the Ecclesiastical Courts." By all indications, Phileas also belonged to no scientific, academic, or commercial associations in London: he took no part "in the sage deliberations of the Royal Institution or the London Institution, the Artisan's Association, or the Institution of Arts and Sciences."
The only organization Phileas was a member of was The Reform Club. At the time of the story, The Reform was a private gentleman's club which admitted no women. Thus, because of his sparse involvement in the political, social, and commercial aspects of London, it was doubted that Phileas was a true Londoner.