Does this man claim to possess a true blessing from Allah Almighty, through which he can heal diseases by touch, or is physical blessing exclusive to the Prophet, peace and blessings be upon him?

If a man is righteous and upright, and by his statement: "He heals all diseases by touch," he means that he strokes thebody of the sick person and supplicates for them or performs legal ruqyah upon them, then he may be truthful. For Allah supports His righteous servant with a miracle, so he heals people by His permission, as happened to the boy mentioned in Sahih Muslim. Stroking with the hand during ruqyah is permissible, and the Sunnah has indicated it, as in the hadith of Uthman ibn Abi al-As, and the hadith of Aisha, may Allah be pleased with her, where she used to stroke the Prophet, peace and blessings be upon him, with his own hand during his illness.