What is the explicit definition about the computablity of non-deterministic Turing machine?
One should not confuse function problems like FNP and FP with the different types of functions computable by deterministic, non-deterministic, alternating, probabilistic, or ... Turing machines.

Instead of starting by looking at functions $f:\Sigma^* \to \Sigma^*$, it is easier to start by looking at functions $f:\Sigma^* \to \mathcal{P}(\Sigma^*)$. A function $f:X\to\mathcal{P}(Y)$ is equivalent with to a subset $R\subset X\times Y$ of the cartesian product of $X$ and $Y$, i.e. to a binary relation.

Since we can assume that the decision problem is already defined, using the decision problem for $R$ gives one reasonable definition for the computable functions (of this form). However, that definition would force the different Turing machines to compute each output-bit separately and independently, which is a small speed penalty. One can allow the different machines to produce the output in a way more suitable to their capabilities, to avoid that penalty.

The more annoying problem is that the form of output is different from the form of input. One could combine that with a stupid translation function $g:S\to \Sigma^*$ for $S\subset \mathcal{P}(\Sigma^*)$ to bring the output into the same form as the input. But if the output was produced in a way more suitable to the different capabilities of the Turing machines, such a stupid translation function might need to be more powerful than desired. Even accepting the small speed penalty would not allow to avoid that problem, since the functions computable by the different Turing machines are often not closed under composition. That problem might be partly solved by using monads (and accepting $\mathcal{P}(\Sigma^*)$ as the form of output), but then the fact that functions computable by deterministic or alternating Turing machines were actually closed under composition gets hidden.