Is there a way to map the concatenation operation over strings to the addition operation over $\mathbb{N}$
Yes, if you don't want an injective mapping: just assign to all strings value 0.
No, if you want an injective mapping:

Empty string must correspond to value $0$. Otherwise,
$$value(\epsilon) = value(\epsilon \cdot \epsilon) = 2 value(\epsilon),$$
- contradiction when $value(\epsilon)\ne 0$.

Let &quot;0&quot; correspond to value $a$ and &quot;1&quot; correspond to value $b$ ($a,b &gt; 0$). Then
$$value(0^b) = b \cdot value(0) = ab = a \cdot value(1) = value(1^a)$$
- contradiction, since both $0^b$ and $1^a$ correspond to the same value.