I also have 6th and a large 1st armored the 1st armored reads fort hood on it.
Would it be common for one person to belong to all of those, or would they be from different people?
The MTS was the Motor Transport Service, and I would guess that was from the same relative who had the Gulf Command patch. The wings and prop would seem Air Corp/Forces related, but I've never seen on like it... for the back of a bomber jacket?(based on size)
I don't know much about the frequency of reassignments a battalion might go through, but I would think that yes one could have been part of one Armored Division and switched to another... particaullarly into the 6th toward the end of the war to beef it up for their plunge into Germany.