Well, prior to the Super Bowl era, the NFL championship was directly equivalent to the NFC championship today. What's more prestigious now? Winning the NFC championship, or winning the Super Bowl?
Those pre-Super Bowl era NFL championships Bart Starr won? He didn't then have to go on and defeat the AFL champion to get the title. The Super Bowl championships Tom Brady has? He had to do both.
If you count only the things that are directly equivalent, Starr had 5 NFC championships and 2 Super Bowl wins. While Brady has 9 AFC championships and 6 Super Bowl wins.
So either way, the same person now holds both records for a Quarterback.
For teams the Packers blew away all the possible competition with 11 pre-Super Bowl NFL championships, while the Patriots and Steelers are tied for the lead with 6 Super Bowl wins.