you don't need to invoke a concept of area to justify this. you can just think of a grid of dots. since multiplication is repeated addition, each row represents one step of the repeated addition.
2x3 = 3 + 3
...
...
3x2 = 2 + 2 + 2
..
..
..
since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.
2x3 = 3 + 3
...
...
3x2 = 2 + 2 + 2
..
..
..
since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.