It makes some sense to refer to Mr. Lincoln's army as serving the union cause, but strictly in an informal way. When making reference to a Federal military unit from a border state, most writers will use the designation 5th Kentucky (U.S.) Cavalry, or 1st Tennessee (U.S.) Artillery rather than attempting to use the word "union" in the title of the command.
From a purely logical standpoint, a writer might decide to pair descriptive words to denote motivation to fight, such as union and secession. Federal and Confederate are appropriate matches for armies raised by the two warring governments, both federal in organization and design. United States and Confederate State of America work equally well.
For the reasons cited, pairing the terms 'Union' and 'Confederate' seem unreasonable, one being a reason to volunteer and fight, the other the name of a government.