I just watched that scene with the first use of "But that's not me/you" again (via the link in your comment) and realized, wow, I got that scrambled. Arya says it, not her father. Her father's still expecting her to fulfill a traditional gender role. At the same time he's also supporting her training in martial arts— something we see from other female POV characters is *rare* in most of the Seven Kingdoms. Because of that I always interpreted this scene not as him limiting her but gently reminding her what society expects.