It is entirely possible that an African American was able to "pass" for white and serve in a white regiment. I sincerly doubt he could have done so openly, particularly considering the racism and bureaucratic barriers of the time. This theory is assuming he had light skin and non-descript features.

We know that Chinese Americans served in "white" regiments, and did so openly, as well as persons from a great diversity of national and cultural backgrounds. Wise may have been assumed to have been of southern European descent by his comrades-in-arms.

Women even got away with serving in regiments, in disguise as men, so you know the examining surgeons generally didn't look too closely at anyone.