what you're saying doesn't make any sense at all though. why would it be any different from when brienne swore to catelyn? brienne has said she doesn't serve the starks she serves lady catelyn.
I mean, it does though… It does make sense.
why would it be any different from when brienne swore to catelyn?
Well primarily because Lady Catelyn was not the head of House Stark at the time, Robb was. A pledge to Cat wouldn’t be the same as swearing fealty to House Stark unless Cat specifically indicated that it should be. And given the circumstances of Robb and Cat’s relationship at the time, it makes some sense that she didn’t. Cat was in trouble with Robb for wanting to save her daughters, Brienne’s vow included saving Cat’s daughters… The two were inherently contradictory. Brienne could not have possibly served Cat and all of House Stark at the same time at that point.
To be clear, I’m not saying that a pledge to Sansa would explicitly mean he wasn’t pledging to her alone, because it could mean that. But given the circumstances, I highly doubt that it would. If Jaime is going to fight for the Starks and the North, he would be pledging to all the Starks. And the only way for him to make THAT kind of pledge would be through Sansa, Lady of Winterfell and current Head of House Stark. It would be either to Sansa or possibly to Jon, depending on how things work out with Jon’s title once he returns.
Swearing himself to Bran though WOULD be a pledge to Bran and Bran alone. Unless Bran specifically indicated that he wanted serving House Stark to be a part of Jaime’s pledge, Jaime would have no duty or obligation to do so, just like Brienne doesn’t.
Swearing fealty to Bran would NOT be the same thing as swearing fealty to the Starks. Swearing fealty to Sansa could, and probably WOULD, be.