So was the president and Ben Gates both Republicans?

Not necessarily. The two main US political parties (Democrat and Republican) have changed dramatically over the course of their histories. The parties today are not what they were in the 1800s. In fact, their values/beliefs have mixed with each other's over time, to the point that the two parties have switched identities on many issues.

Abraham Lincoln and the "old" Republican Party pushed for a more centralized and more involved government, tax-funded and government-controlled programs, and other things that would today be viewed as a liberal or Democrat thing.

On the other hand, the "old" Democrat Party (i.e. in Lincoln's time) was the reactionary party that touted itself as defending old-fashioned and conservative values, saw Lincoln as a dangerous liberal trying to subvert and destroy the American way with his liberal beliefs, and argued for the "keep the government out of my business" motto that has now become standard stuff for today's Republican Party. It also derived much of its origins and support from the American South, and this is a big factor in why the majority of today's South states inherited their conservative political identity (i.e. the modern Republican Party).

To make a long story short, a 2010 Republican does not necessarily share the same beliefs with an 1800s Republican. And a 2010 Democrat does not necessarily share the same beliefs as an 1800s Democrat. In fact, in many ways, they would be opposite from each other. Hope that helps!