Is josiah and joash the same person?

No Joshua son of Nun left Egypt in the Exodus with Moses. (Joshua son of Nun, original name was Hoshea, until Moses changed it)

Joshua son of Jozadak, came years later after the Kingdom of Israel and Judah went into Exile. Joshua son of Jozadak is prophecied to have the same name as Israels future Messiah from the Branch of Jesse and David in Zechariah 6:12