Yes, generally the number on your permit will be the same as what comes on your official state ID/driver's license. However, some states handle this differently. I would contact your state's licensing office to make sure that your specific state doesn't issue two different numbers. I am attaching an FAQ with all the state's Department of Revenues, click on your state to access their DOR website and then search for the licensing division for further information. There should also be contact information on their website; however, you may have to wait till normal business hours to call and speak with someone from the licensing division.
https://ttlc.intuit.com/replies/3302452