I don't think that is true or aligned only to the BBC. Work for any medium to large organisation and you could loose your job if you spout racism in a public forum. At the same time you are free to express any opinion you like, but if it's one that will embarrass your employer such as racism or extremism then you can be summarily sacked.
What has Lineker ever said that he deserves sacking for? He is left leaning but not an extremist. At the same time Andrew Neil was employed by the Beeb for decades and is far more extremist then Lineker but to the right. You could also ask why they would employ a right wing, racist extremist like Johnson to host HIGNFY.
It's clearly an absolute fallacy that people can't have personal opinions or they will get sacked, or that the beeb is left wing agenda driven.