No prisoners do not wear their own clothes.
When a person is arrested and taken to jail they have to change out of their own clothes and change into a set of jail clothes.
The clothes have the jails name on it and indicates that the person is an inmate in that jail.
Then if they go to prison they are taken to jail in another outfit for prison and then when they arrive in prison they are admitted to the prison and processed through.
Then during the arrival at prison they are given prison clothes to wear which can either be a prison Jumpsuit or sweat pants and shirt with the prison name on them.
Then they are given a prison number and wear prison shoes.