Firstly your statement on his apparel is supposition that he was with the poor did not mean he was destitute.
Second.. He was FAMOUS the possessions of FAMOUS people tend to be valued thus no mater what the garments were the Romans would have had a reason to wan them.
Logic is not a blunt tool it takes some skill to us it.
You might want to re-think your logic comment - yours is flawed. Jesus was not famous, as you contend. He was considered a rabble-rouser and criminal. His biggest detractors were not the Romans, but the Sanhedrin who believed that he was trying to usurp their power and authority over religion.
Socrates was, essentially, sentenced to death for defying the authority of the powers that be who also controlled religion. Jesus was, essentially, placed in the same category and crucified as a result.