Yes, very similar. Billy Dixon used a borrowed Model 1874 Sharps rifle chambered in the "Big Fifty", otherwise known as the .50-2.5" or .50-100. Many today believe that the "Big Fifty" was a .50-3.25", or .50-140, but Sharps never chambered that round, nor any round on a 3.25" case. The Brits did, as the ".450-3 1/4" Express", but that was an entirely different round than the American .45-3.25" (which was an early "wildcat" round being chambered by various gunsmiths in their customers' rifles in the late 1880's).
Interestingly, Sharps never referred to their chamberings as .45-70, .45-90, or anything like that. Their reference, rather than powder charge (the second number in those designations), was case length. Their .45-70 was a .45-2.1", their .45-90 was a .45-2.4", and so forth. Theirs was a more accurate way to delineate between the various case lengths of the day, especially when powder charges varied.
A great example is their .45-2.6", their original "Creedmore" match chambering. It held 100 grains of powder and fired a 550 grain bullet. They only offered it for about a year (it is one of the rarest of the original chamberings) before shortening the case to 2.4". It still held 100 grains of powder, still fired a 550 grain bullet, and still had the same overall length. There was just less bullet seated down into the case, for less neck tension, which they found to be an advantage. Both were ".45-100's" by others' nomenclature, but the newer case was .2" shorter. Confusing, unless you use case length instead of powder charge.
The pace of development in those years (1860's-1880's) was pretty remarkable. I mentioned I was shooting 100 grains of powder in my .54 caliber muzzle loader. This drives a patched round ball to that 1,943 fps, but even at .54 caliber, that round ball only weighs 230 grains. Same as a standard hardball load in a .45 auto, just much faster. That was "big medicine" up until about 1865 or so, about as good as we had available for buffalo, elk, brown bear, and other denizens of the West.
Christian Sharps's rifle helped redefine what "big medicine" really was. They were so vastly more effective on big animals than the lowly patched round ball that there really was no comparison. .45 caliber rifles shooting 500 grain and heavier bullets, .50 caliber rifles shooting bullets up to 700 grains (although 450 grains was the most commonly used in the Big Fifty). Especially once ranges increase, the heavy, elongated bullets and their superior ballistic coefficients changed the game entirely.
This photo illustrates what happened. On the far right is a .54 caliber round ball, weighing 230 grains. Left to right are three .45 caliber cartridges, the 2.1", 2.4", and 2.6". The two bullets are 500 and 540 grains respectively. They more or less represent the powder charges that would be used in the .54 muzzle loader as well. But just look at the differences in projectiles - those long, heavy for caliber bullets just "carry" so much better, hit so much harder, and penetrate so much more deeply.
It's no wonder riflemen dropped their round ball muzzle loaders en masse when they saw what these modern breech loading cartridge guns could do. Never mind they were faster to reload, they were so much more effective on game (and man) at so much greater ranges it was immediately "game over" for the muzzle loader.
Billy Dixon simply could not have made that shot with a round ball...