I read somewhere that there are, or were other institutions arguably more ancient. One was the Ethiopian monarchy (ca 975 BC - 1975 AD?) --- but that's gone now, and so "out of the running" so to speak. Another would be Buddhist monasticism, depending on whether or not you see it as unified and continuous enough to be called an institution.
I suppose I am a little more broad minded than that. Gods Church started in the Garden of Eden. Adam and Eve worshiped God there. His Church has gone down through the ages. Abraham. Joseph. Moses. Many leaders and followers. John 4:22 tells us “Salvation is of the Jews.” That was a statement then. It was a prophecy later fulfilled on Calvary. And it is a prophecy to be fulfilled in the future with the latter rain days. God has always had a Church. Jesus just magnified its meaning more clearly. Peace be to you.