Why did the Quran describe the people of Lot as "brothers of Lot" in Surah Qaf, verse 13, while describing the peoples of previous prophets as "the people of" or "the companions of"?

Among the eloquent aspects of the Quran is the diversification of style. Allah has referred to the people of Lut as his people in several verses, as in the Almighty's saying: "The people of Lut denied the messengers," and "And [We sent] Lut when he said to his people." However, in Surah Qaf and Ash-Shu'ara, they are referred to as Lut's brothers, as in His saying: "When their brother Lut said to them, 'Will you not fear Allah?'" This brotherhood has been questioned due to Lut's differing lineage from his people. The answer provided is that Lut married among them and resided with them, so "their brother" was said on this consideration. In the Arabic language, one might say "the brother of such-and-such" to mean his constant companion.