Defining the Inner Product for Product-Hilbert Spaces Let's take a look at what Marinescu & Marinescu say in a particular part of their book "Classical and Quantum Information" on page 11:

Am I right to say, that with this, they also have defined what the inner product is for the hilbert space $\mathcal{H}_{mn}$? Furthermore, is the last sentence a conclusion or rather a definition (because this might be the obvious way to define the inner product on $\mathcal{H}_{mn}$, but not the only way)?
 A: 
Am I right to say, that with this, they also have defined what the inner product is for the Hilbert space $\mathcal{H}_{mn}$?

Yes, their second sentence implicitly defines a inner product on the full Hilbert space $\mathcal{H}_{mn}$. That's because you can use the linearity of the inner product to express the inner product of two arbitrary vectors in $\mathcal{H}_{mn}$ as a polynomial in the two vectors' expansion coefficients in the orthormal basis for $\mathcal{H}_{mn}$.

Furthermore, is the last sentence a conclusion or rather a definition (because this might be the obvious way to define the inner product on $\mathcal{H}_{mn}$, but not the only way)?

It's a definition, but not a completely rigorous one. Exactly as you say, this is a very natural way to define an inner product on the Hilbert space $\mathcal{H}_{mn}$, but not the only possible one; the tensor product is defined to use this particular choice of inner product.
The reason that this definition isn't completely rigorous is that it isn't immediately obvious that the resulting inner product is independent of your choice of orthonormal bases for $\mathcal{H}_m$ and $\mathcal{H}_n$. It turns out that it is, but showing this - and thereby showing that this inner product on the tensor product space $\mathcal{H}_{mn}$ is unique - requires a little bit of extra work.
