Here is a high-brow way to interpret your construction.

Recall that the ideles $\mathbb{A}^\times$ of $\mathbb{Q}$ are defined to be the restricted direct product of $\mathbb{R}^\times$ and $\mathbb{Q}_p^\times$ with respect to the subgroups $\mathbb{Z}_p^\times$.

There is a natural "diagonal" embedding $\mathbb{Q}^\times \subset \mathbb{A}^\times$. Moreover, we can combine all $p$-adic valuations together to get a homomorphism $$\mathbb{A}^\times \to \bigoplus_p \mathbb{Z}.$$

Your "inner product" is then just given by the composition $$\mathbb{Q}^\times \times \mathbb{Q}^\times \subset \mathbb{A}^\times \times \mathbb{A}^\times \to \bigoplus_p \mathbb{Z} \times \bigoplus_p \mathbb{Z} \to \bigoplus_p \mathbb{Z} \to \mathbb{Z},$$ where the penultimate map is component wise addition and the last map is the obvious sum.

How does this help? Well it illustrates how your construction can be put into the bigger picture of the more natural ideles, which are certainly a very important part of number theory.

But I don't think your construction has "any significance, from the perspective of number theory". Number theorists tend to work with the more natural diagonal embedding $\mathbb{Q}^\times \subset \mathbb{A}^\times$ into the ideles, as the ideles have more structure (e.g. form a topological group)

Here is a high-brow way to interpret your construction.

Recall that the ideles $\mathbb{A}^\times$ of $\mathbb{Q}$ are defined to be the restricted direct product of $\mathbb{R}^\times$ and all $\mathbb{Q}_p^\times$ with respect to the subgroups $\mathbb{Z}_p^\times$.

There is a natural "diagonal" embedding $\mathbb{Q}^\times \subset \mathbb{A}^\times$. Moreover, we can combine all $p$-adic valuations together to get a homomorphism $$\mathbb{A}^\times \to \bigoplus_p \mathbb{Z}.$$

Your "inner product" is then just given by the composition $$\mathbb{Q}^\times \times \mathbb{Q}^\times \subset \mathbb{A}^\times \times \mathbb{A}^\times \to \bigoplus_p \mathbb{Z} \times \bigoplus_p \mathbb{Z} \to \bigoplus_p \mathbb{Z} \to \mathbb{Z},$$ where the penultimate map is component wise multiplication and the last map is the obvious sum.

How does this help? Well it illustrates how your construction can be put into the bigger picture of the more natural ideles, which are certainly a very important part of number theory.

But I don't think your construction has "any significance, from the perspective of number theory". Number theorists tend to work with the more natural diagonal embedding $\mathbb{Q}^\times \subset \mathbb{A}^\times$ into the ideles, as the ideles have more structure (e.g. form a topological group)