Well function composition f(g(x)) is not the same as g(f(x)) and when you represent f and g as matrices relative to some suitable set of basis functions then obviously AB and BA should be different. If the multiplication was defined any different, that wouldn’t work.
The way that I used to put this was, "If I put on my shoes before my socks, I'll get a different result than if it I put on my socks before my shoes. Order of operations matters."
