There is an 1865 Census in New York that lists military information - it's very helpful for linking soldiers to their families. But I don't find Martin there. And he's not in the 1890 Veterans Census either.
With so little definite information it will be hard to make a certain ID. We don't know immigration dates or locations, though given the time period (that 1857 date is very helpful) we can guess he came into New York. Most newly immigrated Irish stayed in New York or went nearby.
Monaghan could be spelled in the USA any number of ways. His birthdate isn't exactly known and men didn't always tell the truth about their ages. And we can't rule out that the record
@Toblerone1 found isn't also correct - Martin certainly wouldn't be the first bounty jumper we ever saw.
Even without definite information on him, the record from the NY 73rd fits a lot of the particulars. He's an Irish born laborer about the right age. He was recruited in NYC which fits immigration patterns. And he goes missing at the same battle and about the same date the family is told he died. That kind of persistent family legend usually has a core truth to it. I'd say the family had information from a friend or fellow soldier who knew him well enough to write back to them or the local parish priest with news of his death.
The sticky bit for me is the fact that he didn't die at Gettysburg. The New York muster roll for the 73rd says "Ordered to rejoin his regiment from August Gen'l Hosp. Alex. VA May 17, 1864 but failed to do so." So we know he both survived the battle and was deemed fit enough to return to service. But his records end there and where he went is anyone's guess. Maybe he had a friend send news of his death to the family in Ireland and went west. Or didn't even bother - it's pretty easy to get lost in New York City.
One other possibility that wouldn't be unusual for us to see - he could have enlisted under an assumed name and the death is recorded under that name with no apparent link to Martin Monaghan. For instance, if the record of his enlisting in the US Army and then deserting is correct, Martin might have chosen to reenlist under a different name. He rejoins the army and is killed at Gettysburg under that name. A man who knows him well enough to know his real name alerts the family to Martin's death. But as no one files a pension claim that gives us information between Martin and the assumed name, we don't have that info. Enlisting under an assumed name wasn't that uncommon, it could have happened here.
I realize I'm throwing a lot of stuff at the wall here and we have no way of knowing what sticks. If I had to guess I'd say the man in the 73rd is our guy and he went AWOL from both the Army and his family back home.
@Toblerone1 has your partner ever done DNA testing? If Martin made a new life for himself here he likely had more children and they likely have descendants.