I understand that Yukio did not inherit Satan's blue flames or any demon qualities as Rin did, but still Yukio is a legitimate child of Satan. How come nobody else realizes and/or care about this but when it came to Rin they treated him wrongly?
From a very young age, Yukio trained to become an exorcist which could have helped to boost his reputation. This showed to many that he meant no harm to mankind, but instead to aid in defeating Satan. Also, the fact that he did not inherit the blue flames causes people to perceive him as less of a threat as Rin. I agree that the way Rin was treated was unfair, but the people most likely overreacted towards Rin's blue flames and other demon qualities.