I believe it was a variation on the 1858 Remington.
The original Remington Army had a slot that allowed you to pull the base pin and remove the cylinder for fast reloading at any time. The Army didn't like this feature, since it could theoretically allow the base pin and cylinder to pop out and be lost in combat. Remington modified the design with an unslotted loading lever that blocked the base pin in place. You could still switch cylinders as a method of speedloading, but you had to drop the loading lever first.
Therefore, any Remington 1858 or variant could speed-load by changing cylinders fairly easily. Very easily, if it was one of the first, or had been modified to be like one of the first.
As far as I've been able to find from reading, speedloading with a spare cylinder was never that popular in the cap-and-ball era. In the cartridge era even less so, since with cartridges there wouldn't be as much of a speed advantage (if any) in reloading this way, and carrying cylinders loaded with cartridges wouldn't be practical; the cartridges would tend to fall out. That doesn't mean that somebody wouldn't do it in a Western movie, though. Especially an Eastwood western. His films tended to have less famous firearms in them- Remingtons, Colt cap and balls, the Schofield, Spencer, Savage 99, etc. Whether because he thinks they're cool or because it gives a bit of added authenticity I don't know.