No, that's not a typo. The Mulvaneys were listed as Mulvanys in both the 1860 and 1870 censuses.
This 1860 census lists, numbered family 224, James Mulvany, his wife Bridget, and their 3 sons, Thomas, John, and Patrick. Patrick is listed as 1 year old, which gives him a birthdate somewhere in the vicinity of 1859-1860 - still pretty much in the range we have based on his ages given over the years and the age he was said to be when he died. James gives his age as 35, which means he was born around 1825, and Bridget says she is 28, meaning she was born around 1832. Both parents were born in Ireland, but all three boys were born in NY. James is a Carpenter by trade, and the value of his personal estate is $40. None of the boys - Thomas is, at 5, the oldest - have been to school yet.
What's possibly more interesting is that the family two above them, numbered 222, is the family of John and Ann Mulvany. More Mulvanys, at the same address? I can only imagine that John and James Mulvany were brothers. Possibly they were not - they were cousins, or it was pure coincidence. But brothers is not an unlikely possibility. John Mulvany was 30, and so, being 5 years younger than his probably-brother James, was born around 1830. He's a carpenter, too - might they have worked together? - with his personal estate valued at $50. His wife Ann is 26, and so was probably born around 1834. Both of them, and their oldest son, Michael, 7, were born in Ireland. Their next child, oldest girl Ann, 5, was born in NY, which means that this Mulvany family, at least, immigrated sometime between 1853 and 1855. I can't say whether the same is true for the family of Bridget and James, but given that their 5-year-old, too, was born in NY, I wouldn't be too surprised if the brothers had immigrated together, with their families (although, of course, we don't even know whether James and Bridget were married yet, or whether they married in the US or Ireland). They have another Patrick, this one 4 years old, as well as a two-year-old named Mary. Of all of these cousins, only Michael, 7, has attended school within the past year. I don't doubt, though, that his sister Mary and cousin Thomas would be joining him soon.
It looks like James named his second son after his brother, and the fact that both Mulvany boys named a son Patrick suggests to me that it may have been an important name in the family - or possibly they both had a devotion to St. Patrick, being Irishmen leaving home. But again, we've entered the realm of pure speculation. Does anyone know anything about Patrick's brothers and sisters, or where he grew up? There are no addresses listed in censuses from this early.