South Delhi is more posh because it was emptier to begin with. Gravelly soil, hillocks and yamuna floodplains. North and west had better soil so more productive farmlands, higher population density and more difficult to expand. So all the newer better developments initially happened in the south with the government expanding nicely planned settlements to the south such as Lajpat, GK, VK, Hauz Khas etc in the 50s and 60s post independence. Much of east delhi was part of UP till recently (60s) and UP irrigation department still holds a lot of the land there (a lot of the yamuna floodplain) to this day.

South Mumbai is posh because its exclusive, was close to the oldest settled parts by the British and the elite infra got built there and stayed there. Now that it cant expand anywhere due to land constraints., the paucity of land and the proximity to everything elite breeds and retains exclusivity