This is closely connected with the royal ancestry question. Statistically, all presidents descended from Charlemagne (which is all of them) are cousins of all the others. Genealogically, all presidents with traceable Charlemagne lines are traceable cousins of all the other presidents with traceable Charlemagne lines (and you don't need to go all the way back).
With the early presidents, that's pretty much the whole story. People like Washington, Jefferson and John Quincy Adams don't share immigrants, and they don't show cousinships with presidents who don't show Charlemagne lines.
Recent presidents are a bit different. Bush has about 10 presidential cousins with an immigrant as common ancestor, and a couple with a common ancestor born in America.
(All in the Wikitreeverse, ignoring any possible divergences from other versions of reality)