Is our saying "may Allah bless him and grant him peace" after mentioning the name of the Messenger considered a prayer upon him from us, or does it fall under His Almighty saying, "Make not the calling of the Messenger among you as your calling one of another"?

The Almighty's saying: "Make not the calling of the Messenger among you as your calling one of another" does not urge prayer upon the Prophet, but rather prohibits addressing him by his mere name, out of reverence and veneration for him. So they would say: "O Prophet of Allah, O Messenger of Allah." As for the urging of prayer upon the Prophet, it has been indicated by other proofs, and its meaning is to pray for him with more blessings and goodness.