@Nathan Stuart Why do you question that Arthur is his brother? It looks very clear to me from Census and military records.
lupaglupa. Yes, it was merely a doubt raised, not a conclusive determination, from bits of incomplete and inconsistent information viewed.
From various online sources, like
https://www.historytools.org/people/emery-hamilton-biography-history-and-inventions , it's claimed that in Apr. '61, Emery and his brother, Arthur, 'joined the US Army of volunteers to serve in the American Civil War.' Although this is a broad statement, it seems reasonable to therefore believe (if they were brothers), that both would have enrolled and continued together in the same WI unit. But later, none of the rosters of the WI infantry units present at Antietam (i.e. 2, 3, 5, 6 & 7 WI) show Emery's name. Only Cpl. Arthur Hamilton appears in Co. H, 2 WI. If Emery did serve at Antietam, as suggested was a possibility in the OP, could he have belonged to another unit, say a NY unit (he was born in NY), or a non-combat specialist unit, like in engineering or communications, at this time. The first recorded military service found for Emery was as a Major in the 'First Regiment of Engineers Corps D'Afrique', a unit that was not organized until Apr. 28, '63. If Emery did not serve in the same unit as Arthur, and perhaps was not present at Antietam, thought this might cast on these various other online claims made that they were brothers.
Furthermore. In the relevant 'Find-a-Grave' listing for Arthur Hamilton (see
https://www.findagrave.com/memorial/42294539/arthur-t-hamilton ) it shows his only sibling was "Mary Lavantia Goodrich (nee Hamilton)' – there is no mention of another sibling on this site. However, the Find-a-Grave listing description for Mary (see
Mary Lavantia Hamilton Goodrich (1836-1922) - Find a Grave Memorial ), who died in 1922, states, … She had two brothers, one of whom Arthur, was killed in action in the Civil War…….The other brother died in New York City one year ago.'… There is no mention here of the other brother by name, but it's quite possibly Emery.
In view of these findings, and what you say you found from the military and census records, can now accept that it appears likely that Emery and Arthur were in fact brothers, although Emery's service period may not be fully accounted for.