There was never any kind of “rule” like this. There were bishops of common birth and ordinary priests of noble birth who never became bishops.
Sometimes, yes, there were political pressures brought to bear for appointment of bishops whose families were allied with the powerful. But that was never part of any “rule”, but was a corruption in a pretty primitive time.
Having said that, it probably was of some advantage in becoming a bishop to have come from a noble or at least well-to-do merchant family, because your underlying education might have been better that that of most other priests. One of the Benedictine reforms, in fact, was to improve the education of priests from what it had been. Being a bishop was no easy task then. It isn’t now, but the challenges are different in many ways.