Uchoka pretending to be Italian may seem arbitrary, except for the fact Italian and French people have a rather close relationship (Italians even refer to the French as "our cousins from beyond the Alps"): with Mary being half-French, Uchoka's act of being Italian makes more sense.
At Aiichi, the place is more disciplined and almost all trained fighters are all swordfighters or baton users trained in Kenjutsu practitioners (in the entire student body there's a grand total of six who aren't Kenjutsu fighters using swords or batons, and Nomura and Amou are the new guys), while Hokkai is a place where fights and duels happen often and the fighters tend to use more styles, including polearms, rare ones such as Jukendo and various types of foreign fighting styles. It's likely that the discipline imposed by the Five Supreme Swords at Aiichi stabilized the situation in favor of Kenjutsu and other styles only appear when someone with previous training outside it enrolls, while at Hokkai the constant fighting caused everyone to try and find something that could counter or surprise potential opponents and look into rare and foreign styles and other kinds of weapons.
After their duel over the right to discipline Nomura, Mary tells Rin that today she's been the most skilled, and during Hokkai's invasion Rin shows familiarity with HEMA techniques that literally nobody else but Mary and Uchoka at Aiichi had. Mary and Rin are in the same year, they likely competed for years over whatever open spot in the Five Supreme Swords there would be when they got in high school - leading to numerous duels and Rin becoming familiar with European styles.
Amou is the only female student to wear a jacket in late spring and early summer. Considering she's a traditionally trained Uechi-Ryu Karateka she likely grew up in Okinawa, and in the Chubu region she's simply cold.
