Exactly - hence both Kew Bridge and Barnes Bridge are perfectly sensible station names. As is Putney Bridge!
Putney Bridge station was originally Putney Bridge (Fulham), then became Putney Bridge and Hurlingham in 1902, and lost its suffix in 1932. It is at the end of Fulham High Street, in the original settlement of Fulham. What people now call Fulham is really Walham Green, which is what Fulham Broadway station was called until 1952. Walham Green and Hurlingham are place-names that have more or less disappeared from public consciousness, mainly because there now aren't stations of that name (or bus terminals: when Sands End, where Fulham power station used to be, became the terminus for the 371 bus, this revived an almost forgotten place-name).*
Kew Bridge could have been called New Brentford, as it is fairly close to that part of the town, or maybe East Brentford or Brentford East, or perhaps Strand on the Green, which is downstream a bit from the bridge. Barnes Bridge station is closer to the centre of Barnes than Barnes station, but I suspect that as it was on the Hounslow loop the LSWR felt that it would be best to keep the original station as simply Barnes, as it is on the main line, despite its being a fair old walk across the common from the centre.
* Apropos forgotten place-names, I intend to do a Trivia posting on places in London and the surrounding area which have become forgotten as they either never had a station named after them or don't have one any more. I need to do a bit more map-studying before I submit it.