I think the greater significance of Grant's statement is that Bern thought he was seriously teaching/training her and has built much of her life, views and dreams around Grant's lessons. But Grant has just informed her that it was only a game as far as he was concerned and he did not intend for her to take the lessons seriously into adult-hood. Quite possibly, he even held the same views against female knights at the time but Bern believed he seriously gave her the hope and goal to overcome that and one-day join the knights.
I always thought that Grant was supposed to have the same streak of white hair which is why Orransong spoke of the resemblance between the two here: http://flipside.keenspot.com/comic.php?i=394
AHHH! Now I recall, it was a response from Brion to a question I made about it. guess he's changed his mind since *shrug*. There still seems to be a certain level of resemblance in build and hair structure.
I'm not sure it was neccesaraly a game to him, its just that he only taught the split-rose technique as Bern was only a child at the time.
It might well be that Grant did intend to teach Bern some adult fighting styles as she got older, but then the brothel burnt down and he believed she died in the fire.
Ofc that does not mean that he intended for Bern to join the knights when she was older, it is possible that Grant shared the belief that only men could join the knights, and was slimply teaching Bern some swordfighting skills so she would be able to defend herself if needed.
RE: Bern's white patch of hair.
Since there has been no mention of her mother having a smilar patch of white, and Grant has no white patch, it's possible that Bern didn't have it growing up and it turned white as a result of the shock of the brothel burning down and killing nearly everyone she knew (i'm sure there was a page which showed Grant training Bern as a child, but i cannot remember if she was shown having a whitespot at the time ). That might explain why Grant didn't recognise her when he opened the door.