You could try to sand and refinish the floor, but the stain may be too deep to sand out. I know it may not be the prefered solution, but I have spoken to someone at work that was a professional grain and stain restoration specialist. He can take a piece of grainless composite, and paint, stain, seal, clear it to match perfectly to any preexisting floor. Additionally Ive been hearing of a chemical you can get that will remove pet stains in wood. It supposedly attaches to the stain on a molecular level and evaporates it out, but I dont know what its called.
Maybe you could find someone like that in your area. I also saw that done on the TVshow ''This old house.'' Its possible you could get some information from thier website or email them for matching the old floor, and I mean ''old'' in a good way. 1890 is deffinitly something to try and preserve...