Is our talk about the manager who mistreats us considered backbiting, even if it is for the purpose of relieving ourselves?

Gheebah (backbiting) is your mention of your brother with what he dislikes. Your mention of the director with what he dislikes, such as his shortcomings and his superior's reprimand of him, falls under the prohibited gheebah. The Sharia (definition) mistakes he commits do not permit backbiting him; rather, you should advise him. If his management causes disruption, then the higher management can be addressed. Backbiting him does not benefit the work or the backbiter; rather, the backbiter's good deeds may be transferred to him, or his bad deeds may be transferred to the backbiter.