The NPS site has a couple men with that name who fought. None from New Jersey though. These men fought from Ohio, Indiana, Illinois, and New York.
The man from New York died in 1913 in the Soldiers' Home in Dayton, Ohio, and his body was returned to Jamestown, New York (indicating that he could have lived there originally). His pension index card makes no mention of a widow's pension that was applied for, so there is no mention of a widow's name. He had a son named Walter.
The New York Thomas Tansey was born 1838, according to his gravestone. His papers at the Dayton Soldier's Home says that he was born in New York. He is accounted for there in 1909, but your Thomas seems to be in New Jersey in 1910.
Your Thomas, according to census records and his draft registration, was born 1833-1835.
I checked the 1890 Veterans' Census for Monmouth County, New Jersey, and there's no one listed with that name.
The 1910 census asked if people were veterans of the Civil War. It looks like Thomas was still alive in 1910, but his entry has no notation that says that he was in the war.
None of this completely rules out that your Thomas was in the war. It's possible that he moved around and that he didn't report his service in the census later in life. However, it seems most likely that he was not in the war, based on all that was found on him.
Did you have something that indicated that he did fight in the war? Or were you thinking it was a possibility based on his age?