He, sallallaahu ‘alayhi wa sallam, would not be biased in religious affairs, even though they were his relatives:

Ibn Hajar said: “They said regarding Al-’Abbaas “the son of our sister” because they are the maternal uncles of his father, ‘Abd Al-Muttalib, for the mother of ‘Abd Al-Muttalib was from Madinah. Her name was Salma bint ‘Amr ibn Uhayhah, and she was from the tribe of An-Najjaar.

The reason they called ‘Abbaas, “the son of our sister” is so that it would seem that they were doing the favor of setting him free because of his relation to them. They did not say, “Your uncle,” so that it would not seem that they were doing a favor upon the Prophet of Allah, sallallaahu ‘alayhi wa sallam. This choice of wording shows their intelligence and good manners in speech.”

Ibn Hajar also said: “Ibn ‘Aa’ith narrated that when ‘Umar was put in charge of restraining the captives, he tightened the restraints of Al-’Abbaas. When the Prophet of Allah, sallallaahu ‘alayhi wa sallam, heard him moaning out of pain, he, sallallaahu ‘alayhi wa sallam, was unable to sleep. When the Ansaar heard about that, they released him from the ropes. When they realized the Prophet of Allah, sallallaahu ‘alayhi wa sallam, was pleased with their action, they wanted to set Al-’Abbaas free with no ransom to increase his pleasure, but the Prophet of Allah, sallallaahu ‘alayhi wa sallam, did not allow that.

Ibn Hajar further said: “The reason he did not allow them to do it is so there would be no bias or favoritism in the religion.”