He carries that blood as much as she does, though she doesn't know that yet, and he hasn't decided how to tell her. More than just carrying that blood, he's also a walking reminder of why Robert's Rebellion happened in the first place. He doesn't know how things were between the Starks and the Targaryens before Rhaegar and Lyanna met, but he knows what the stories say now. At least on the Stark side. He's a walking reminder of that, too.

"That doesn't mean that everyone else was." He knows the story about Rhaegar isn't true, but he doesn't know what is true. And while he is trying to reconcile things Sam told him about Danaerys with his own feelings, he also realizes that the one he's talking to hasn't done those things yet.

He knows what it means to be a Stark. He has no idea what it means to be a Targaryen.