Is swearing on the Qur'an to abandon forbidden acts, by saying, "By Allah, this mobile phone is forbidden to me if I commit this sin," considered an oath or a vow, and can an oath attain the status of a vow without the person being aware of the rulings of vows?

Your statement: "By Allah, this mobile phone is forbidden to me if I commit this sin" is considered an oath (yamin) and not a vow (nadhr), because you did not commit to an act of worship. If you return to the sin again, you owe an expiation for breaking an oath (kaffarat yamin), and the mobile phone does not become forbidden to you. You must repent to Allah and take the necessary steps that will help you avoid sins.