Doc;
Both Sharps and Remington offered bayonets for their respective military rifles, but I'm not sure exactly who made them. I had always assumed that it was an "in-house" kind of deal, but it's possible that they were farmed out to one of the major New England cutlery companies like Ames. In any event, if the rifle was originally sold to a military organization, it would have come complete with bayonet, as supplied by the manufacturer. So if your bayonet fits like it was made for it, it probably was!
BTW, there are a LOT of socket bayonets out there that were made in the late-19th Century which look a LOT like the standard Springfield bayonets. The Martini-Henry, the Peabody-Martini, the Remington Rolling Block and several other military rifles were fitted with bayonets of a surprisingly universal form. However, since the muzzle dimensions of all of them were just slightly different, while most would "fit" on one another, only the purpose built bayonets would fit well. So I'm going to make the guess that you have the purpose-built bayonet to go with the rifle, especially if the wear and patina are pretty much the same on both.
Hopefully I can grab a chance to swing by tomorrow and see it!
Cheers!
Gordon