Is there any way to define arithmetical multiplication as other thing than repeated addition? Is there any way to define arithmetical multiplication as other thing than repeated addition?
For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b \text{-times}}$ or $\underbrace{b+b+\cdots+b}_{a \text{-times}}$?
Given two sets $A$ and $B$ of cardinality $a$ and  $b$, respectively, the cardinality of the cartesian product $A\times B$ is called the product of $a$ and $b$, and is denoted by $a\cdot b$.

Update

When I wrote this answer I didn't have infinite sets in mind. I just wanted to convey a mental picture of multiplication that does not involve repeated addition.
No. Because this is the fundamental definition of arithmetical multiplication, that, repeated addition is known as multiplication.

If you try to generate another definition of multiplication then you have to use this definition either directly or indirectly.