The machimoi were soldiers of native Egyptian heritage during the Ptolemaic period, when Egypt was under the control of an originally Macedonian dynasty. They would have occupied a lower “auxiliary” position in the army and served under Greco-Macedonian commanders, whereas soldiers of Greek and Macedonian heritage occupied the elite ranks. It was a bit like how African-American soldiers in the Civil War tended to fight under white commanders.
I believe the upper garment my machimoi is wearing is supposed to be padded linen. You may recognize it from certain computer games like Rome II: Total War, although truthfully I haven’t ever seen it depicted in Egyptian art. I presume it would have been a recent development after various foreign conquests like those of the Ptolemies.