All I have is what's on her marriage certificate where she listed her parents as "Matthew Auchmuty" and "Elizabeth Thompson" of which there is a couple of that name married in 1827 on Longfordancestry.com. They list Matthew as the son of John, and the GGGG Grandson of Arthur Auchmuty who first settled in Longford. (I don't know how much of the site is speculative, but is seems well-sourced to me.)
As Fannie Elizabeth Auchmuty was born on April 15, 1850 in NYC, she and Eshelda are probably the same. I've never run across that given name however. She put Elizabeth on her marriage certificate, but there is some question as to her middle name - I've also run across the name "Zedildach" in old family papers.
What I know of Fannie is that she was a governess to a family called Mendoza in Cuba at some point, and that she had contact with Richard Tylden Auchmuty. This I know from family lore, which says that Fannie's family ran a station on the underground railroad. This predates any knowledge in our family of Fannie's origins, all of which I've uncovered. I don't believe that Fannie ran such a station - it doesn't fit with anything else I know - but RTA definitely did. Thus, I surmise they had some contact, but since they traveled in very different social circles from what I can tell, probably not much. Until reading the post on this forum, I had no idea she had any siblings at all.
The only other Matthew I've found is an Auchmoody from New Paltz. He was in NYC during the Civil War, but I'm not sure when he arrived. For a while, I thought he might be her father, but it was always an awkward fit. According to the Huguenot Historical Society, he never married and had no known children. Longford Matthew is the only one who fits, not to mention both mother and father match what Fannie provided herself.
One last thing. I recall reading once that there was a Caroline or Catherine or similarly named Auchmuty living in NYC around the time of the 1880 census. I don't know anything else about her.