How could the Prophet (PBUH) become angry with someone who shames a Muslim, while the Holy Quran shamed Al-Walid ibn Al-Mughirah by Allah's saying: "Cruel, moreover, and an illegitimate pretender"?

The aforementioned verse is not merely an insult or a taunt directed at a specific individual; rather, it is a prohibition for the Prophet Muhammad, peace and blessings be upon him, and his nation from obeying someone who has reached the utmost limit of disbelief and evil, namely the "zanīm," who is known for wickedness and is often an imposter and illegitimate.

There is no contradiction between the verse and the Hadith. Mentioning a disbeliever by his blameworthy characteristics to warn people against him and to avoid obeying him is not a taunt. Similarly, mentioning a wicked person by his wickedness to warn people against him is not backbiting.

Insulting and taunting a Muslim are characteristics of the Age of Ignorance (Jahiliyyah).