There's no writing that I can find in either HoME or Letters bearing on this question. A few mentions of family names, general discussion of the names themselves, but nothing relating to why Hobbits (and, presumably, Bree-folk) have this particular distinction.
Perhaps the closest is Letter 25, from which I'll quote:
The dwarf-names, and the wizard's, are from the Elder Edda. The hobbit names from Obvious Sources proper to their kind. The full list of their wealthier families is: Baggins, Boffin, Bolger, Bracegirdle, Brandybuck, Burrowes, Chubb, Grubb, Hornblower, Proudfoot, Sackville, and Took. The dragon bears as name – a pseudonym – the past tense of the primitive Germanic verb Smugan, to squeeze through a hole: a low philological jest. The rest of the names are of the Ancient and Elvish World, and have not been modernised.
It's probably worth considering the narrative structure of both the Hobbit and LotR. In both cases, Hobbits are intended to be a "touchstone" for the reader, and both books start with the relative familiarity of the Shire. We view the main action through their eyes, and as the story unfolds it passes from comfortable and familiar surroundings into the world of myth. Tolkien himself was aware of this function of Hobbits, as he notes in Letter 182 (discussing the Silmarillion):
Though I do not think it would have the appeal of the L. R. – no hobbits ! Full of mythology, and elvishness, and all that 'heigh stile' (as Chaucer might say), which has been so little to the taste of many reviewers.
Christopher Tolkien also notes the same in his foreword to the Lost Tales: "in The Silmarillion (there is) no 'mediation' of the kind provided by the Hobbits".
It's worth noting that Bree-folk were mentioned in the comments to the question, and I touched on them above; we should remember that they also have family names. Barliman Butterbur, Bill Ferny, Harry Goatleaf. As Tolkien notes in FotR (At the Sign of the Prancing Pony):
The Men of Bree seemed all to have rather botanical (and to the Shire-folk rather odd) names, like Rushlight, Goatleaf, Heathertoes, Appledore, Thistlewool and Ferny (not to mention Butterbur).
This supports the thesis that names also reflect the passage from familiar territory to the mythical world. Here we see that in Bree we have the use of family names for familiarity, but the family names used seem odd to the Hobbits. They're leaving their familiar territory, and soon enough they'll be out in the Wild. It's interesting to observe that Aragorn (in his "Strider" guise) - something of a pariah in Bree - has no family name.
In the light of this, it's probably not surprising that the Shire and it's environs were where family names were used, thereby providing another element of initial familiarity.