The family name is Montresor. This is shown unmistakably when Montresor and Fortunato are down in the catacombs.
“These vaults,” he said, “are extensive.”
“The Montresors,” I replied, “were a great and numerous family.”
Montresor is obviously a French name. Poe does not explain why Montresor should be living in Italy. It would appear that the Montresor family could not have resided in Venice for more than a few generations. All those bones in the catacombs are not bones of Montresors--and Fortunato knows it. He regards Montresor as a Johnny-come-lately, and some of his "thousand injuries" may have been disingenuous questions such as this:
“I forget your arms.”
He probably is sure Montresor's family has no coat of arms. The fantastic coat of arms Montresor describes for him is undoubtedly pure invention.
“A huge human foot d'or, in a field azure; the foot crushes a serpent rampant whose fangs are imbedded in the heel.”
Fortunato is heavily intoxicated and doesn't realize he is being mocked. Poe takes pains to emphasize that Montresor is French. Montresor gives Fortunato two French wines. He uses such French terms as flambeaux, rapier, and roquelaire. Poe's story was published in 1846. Fortunato had supposedly been dead for at least fifty years, which would mean that Montresor had been living in Venice since 1794. Montresor may have fled from France because of the French Revolution and its aftermath. He may have been the only Montresor to come to Italy. Which would mean that few if any of the bones in the catacombs would have been those of his family members. He appears to be the only survivor of a small family, including a wife and perhaps a few children. The fact that he lives in a palazzo means nothing. He could be renting it cheaply because those old mansions became white elephants after the original families died off. All the bones are probably those of the original owners' families. Montresor had to take the bones along with the palazzo because there was no way to dispose of them. In a city like Venice it might have been impossible to bury people because there would be water just a few feet down. They have the same problem in New Orleans; they bury corpses above the ground.
It seems that Poe had a preference for old, dilapidated dwellings. In "The Murders in the Rue Morgue" his narrator describes the place he shared with C. Auguste Dupin:
It was at length arranged that we should live together during my stay in the city; and as my worldly circumstances were somewhat less embarrassed than his own, I was permitted to be at the expense of renting, and furnishing in a style which suited the rather fantastic gloom of our common temper, a time-eaten and grotesque mansion, long deserted through superstitions into which we did not inquire, and tottering to its fall in a retired and desolate portion of the Faubourg St. Germain.
And in Poe's morbid tale "The Fall of the House of Usher," the whole building collapses shortly after the narrator comes to visit his old friend Roderick Usher. The story ends with these words:
...my brain reeled as I saw the mighty walls rushing asunder—there was a long tumultuous shouting sound like the voice of a thousand waters—and the deep and dank tarn at my feet closed sullenly and silently over the fragments of the “HOUSE OF USHER.”