Because it fit when the classification originated and has never asked to be reclassified. South Korea was also considered a developing country and was cast as the global South back in the early days, but diplomatically and through its membership in the OECD became a developed/Global North country. If Singapore is offended it can ask UNCTAD and get reassigned.

Ultimately it largely doesn't even matter. It's a casual shorthand that in the overwhelming majority of cases is an accurate split between developed and developing/poorer countries. Some tiny city-state counterpoint isn't really convincing. Orgs like UNCTAD use it to high-level report on progress in lifting up developing nations.

As to alliances, ignoring that you're completely backtracking on your original post regarding that (you know the one where Australia is actual pals with all its neighbours and the N/S thing is a big lie), for obvious reasons the world's most prosperous countries tend to have common interests. Not to mention that a number of countries with a shared history (e.g. the commonwealth and the colonies) ended up being some of the richest countries.