Not directly, but here are some theories:
We learn in Spectre that Bond, after his parents' deaths, was adopted by another family. Perhaps that family was more working class? I don't remember the specifics.
We also know that Bond loathes Skyfall (the estate, not the film, which presumably - if he saw it - would give him a horrible unease to the point where he wouldn't "like" or "dislike" the movie, but wonder what the ramifications are for a perfect documentary of his life). Sorry, I got side-tracked. He hates his home. He didn't like it. It's possible that Vesper's guess is incorrect, but he prefers her thinking he's working class rather than asking questions about his background, questions he doesn't like the answers to. His pause after she guess "correctly" - in this scenario - indicates that he's contemplating telling her or not telling her for a moment and chooses not to (or the moment passes).
Those would be my best guesses for continuity. I kind of like the second because it actually builds depth for the character rather than shuns it. I guess Vesper didn't look up the Bond name before going on the assignment, though, since she would have certainly found all of that information about Skyfall and the Bonds with a quick internet search.