This method is also referred to as Mayan Multiplication which isn't necessarily exclusive to Japan or a technique the Japanese invented. It's not Mayan either, but just a neat technique to make multiplying numbers easier. Although it might be hard to perform in your head this way.
The babylonians, egyptians, and romans had their own techniques because multiplication by hand is not easy, and it's time consuming. Techniques like these enabled those civilizations to build wonderful engineering marvels. The point is these techniques don't violate math or subvert teaching math because it's not the same thing.
This subject was originally called arithmetic when I was in school which was different than math. Arithmetic was the rudimentary techniques for how numbers were added/subtracted/multiplied/divided where math was word problems that required you know arithmetic to solve but also required logic and abstraction. So yes arithmetic != math, but that doesn't mean teaching arithmetic is NOT useful. They are different concepts that are related, but not the same.
I think this is perfectly fine to teach someone this. The whole computation isn't math argument is a misplaced. We're talking about 2nd or 3rd grade students. They'll learn math at the higher levels again.