Is it permissible to say that we are servants of the Prophet (PBUH) or that we serve him, or is this exclusive to Allah the Almighty?

A servant is one who attends to his master to serve him diligently. The word "service" (الخدمة) has not been used in the Quran or Sunnah (definition) to describe the worship of God. However, scholars have applied it to obedience and worship of God, such as Ibn al-Qayyim, who narrated from the Salaf that "whoever is pleased with the service of God, all things will be pleased with his service." As for the phrase "servant of the Messenger," it has a valid meaning, in the sense of one who is obedient and follows him. However, it is advisable to avoid this phrase because it is not mentioned in the sacred texts, nor is there any evidence of scholars using it in reference to the Prophet, peace and blessings be upon him, and to avoid inappropriate claims. Expressing oneself with terms from the sacred texts is more appropriate. As for describing a scholar engaged in Hadith (definition) as a "servant of the Sunnah," this is common and carries no harm.