I was doing g a bit of reading the other day and I read that Constantinople was taken by the Ottomans just under 500 years ago, and from then on, it was known as Istanbul

"Istanbul" is a contraction of a Greek phrase that means "in the City," which was a colloquial way to refer to Constantinople. Officially its name remained Constantinople (in Arabic, Kostantiniyye) until 1923.

and was (and still is) a primarily Muslim populace.

Actually the city was repopulated in a cosmopolitan fashion after the Ottoman conquest, and it stayed that way until after World War I. Mehmed II let those who fled return and ordered people from all over the empire to come and settle in the city. I talk about it at length here.

How did eastern Orthodox religion maintain its leadership from the "Patriarch of Constantinople" up to today?

Mehmed II and the Ottomans in general were tolerant of different religions and in fact used their religious leaders to govern them. The patriarchate was vacant at the time of the conquest due to doctrinal conflicts. Earlier in the 15th century, there was a hamfisted attempt to force a union of the Catholic and Orthodox churches (at Orthodox expense, doctrinally) in exchange for military aid that never came. It was official imperial policy that the union was in effect, but it so alienated the native clergy that they had trouble finding anyone willing to be patriarch.

By appointing a new patriarch (and opponent of the union), Mehmed II cemented his role as inheriting those of the Emperor and rule over Orthodox people.

How did such a legacy live on when the Byzantines fell and their city converted?

In so many ways! Probably the most obvious legacy is in Ottoman architecture, which adopted Byzantine forms as a model and developed it further. See the Byzantine Hagia Sophia and Ottoman Blue Mosque.