I am not sure I quite follow where you registered your name wrongly, but I would surmise that when you created your account you accidentally switched first and last name? Which means if you put it correctly when trying to book a ticket you get an error message?

You do need to get through to AMC to get it fixed, they will probably ask for evidence of your name to change it. Which will end up being bit cumbersome. But if you already have miles in your account, there is probably no way round the hassle.

While it is quite frustrating, I would say that the first and last name fields are pretty clearly labelled. Admittedly, we all make mistakes sometimes, but it is not really ANA's fault.