Pope is indeed a cultural term meaning father. The Patriarch of Alexandria also refers to himself as "Pope".
No, the One, Holy, Catholic and Apostolic Church refers to him as Pope, and gave him that title long before the bishop of Rome took it upon himself (it is theorized that the usage migrated from Egypt to Africa and thence to Rome).
The official titles of the Pope are: Bishop of Rome, Vicar of Jesus Christ, Successor of the Prince of the Apostles, Supreme Pontiff of the Universal Church, Primate of Italy, Archbishop and Metropolitan of the Roman Province, Sovereign of the State of Vatican City, Servant of the Servants of God
The official titles of the Pope are "the Pope and Patriarch of the Great City of Alexandria, Libya, Pentapolis, Ethiopia, of All Egypt, and All Africa, Father of Fathers, Shepherd of Shepherds, Highpriest of Highpriests, thirteenth of the Apostles, and Judge of the World."