I think you're right it's not a Doggets Coat and Badge, but, from Wikipedia:
The winner's prize is a traditional watermen's red coat with a silver badge added.
I would guess that what we have here is a "traditional watermens's red coat", and so presumably this is an eighteenth/nineteenth century member of the Company, with a different badge on the arm. The badge looks like a version of the Company's crest - see the one on the top of the page here with the curved fish as a supporter and a crest on top.
(Note that as well as the scene clearly being of London as seen by St Paul's and the bridge, the river has two or three boats shown)