The Queen does not own a passport, despite making numerous official visits overseas. As you point out, because the British passport is issued in the name of the Sovereign, it is not necessary for The Queen herself to have one.
Although The Queen does not require a driving licence by law, she learned to drive in 1945, while still Princess Elizabeth, as a member of the Auxiliary Territorial Service. She would have required a licence at this time, since she was not the Sovereign and would therefore have been required to hold a driving licence under the existing acts of that time.
In a similar fashion, The Prince of Wales currently requires both a driving license and a passport. On ascending the throne, he would have no legal need for such documents, but he would still nevertheless possess them.