He is obviously referring to the US.

As part of his political theory of the concurrent majority, John C. Calhoun suggested that a dual presidency may work in regards to power sharing between minority and majority interests.

His agenda rested solely in protecting Southern political interests, so it can be surmised that these two presidents would have represented the North and South broadly.

The overall popularity of this idea amongst contemporaries I do not know, but I am not familiar with any popular movement to implement this form of power sharing. You could argue that Calhoun's rhetoric helped create Southern nationalism and thus the Confederacy, but they were clearly interested in independence, not a dual executive.

In South Africa when the Apartheid system was coming undone, Calhoun's theory of the concurrent majority was somewhat resurrected as a potential way to balance political power between whites and blacks. But it really did not go anywhere that time either.