That's actually how, up until about the late 20th century/early 21st century, the Gaels in the Scottish Highlands/Islands would distinguish each-other. For instance, on Lewis, there is an almost infinite number of MacLeods, so naming your son Donald MacLeod is like naming him John Smith.
To get around this, Gaels would do exactly as Askot says (and sometimes still do): they'd name Donald MacLeod Domhnaill Dubh - dark haired Donald - if he had black hair, or Domhnaill Ban if he had blonde hair. They could also call him Dohmnaill Mor nan Orain if he was into music/song writing - Big Donald of the Songs, or maybe even Domhnaill Mhaira if he was a sailor - Donald of the Sea. You would never say "Oh, do you know Donald MacLeod?" because there would be about 20 in your 10-20 mile radius at least. The other way they'd do it is say "Oh, do you know Donald Archie Ian Gillespeag?" Donald, son of Archie, Ian and Gillespeag. A kind of patronymic. It used to be frowned upon if you couldn't go back 7 generations.
Perhaps Dwarves could have a clan/family name that goes back to the age of myth, but also a name that they would earn for a deed they have done e.g. "Mebzuth Shieldbroken" because he broke a famous warrior's shield in battle, but his clan name would be Mebzuth Oiledpulleys because of some bizarre notion in Armok's head when he was hammering the clan progenitor's head into shape on his holy anvil. An introduction might go as follows: "Hail traveller, I am Mebzuth Shieldbroken of the clan Oiledpulleys, son of Ingish Goblincleaver, son of Dastot Burningrages, son of Vabok Holybearded, son of Geshud Windowlicker."
An alternative name may be "House". So instead of "the clan Oiledpulleys" it becomes "Mebzuth Shieldbroken of the House of Oiledpulleys".