Sansa blushes a bit at that because she suspects he means caring in a way that isn't simply friendship and perhaps something more and she has no idea what to do with that. In Westeros, courting is rare and something she never got a chance to experience even though she desperately wanted to.

"So you intend to court me? Is that what you mean?"

Sansa thinks it isn't so formal here and certainly not from the world Chris describes but she needs a word to put to it.