Perhaps, but the jacket would then have small holes in the lapel, well that is if it were pinned on. Sewing it on would cause issues as well. Both men may have purchased these to wear home on leave or for the photograph. Note the artillery corporal wear a second (identification ?) badge as well.
The top image is from a carte de visite of George A. Heimbach, Battery F, 1st Pennsylvania Light Artillery, 2nd Army Corps. It was published in An Unusual Shoulder (Lapel?) Patch Insignia from the Civil War, by Edward Steer, Jr. which was published in the Journal of The Company of Military Historians , Spring 1980 issue.
In an attempt clean up my personal library I have been going through old magazines in an attempt to get rid of some. I will not be able to bring myself to part with the old Journal of The Company of Military Historians issues. Sadly they do take up a fair amount of space, my library room is fairly large and I can make room for my issues of the Journal of The Company of Military Historians