Is the equivalent function between isomorphic domains called a dual or something else? If you can also provide citations or examples of mathematicians using the terminology then that would be great.
For example is this the correct use of terminology? Or do I have to change the codomain to a vector space as well?
Let $f(x,y,z)=xe^z+ln(y)$. This has domain $\{(a,b,c)∈R^3 ∶ b≥0\}⊂R^3$. Find the equivalent function, $f^*$ from a subset of a vector space. Define $f^*(〈x,y,z〉)=xe^z+ln(y)$, with domain $\{〈a,b,c〉∈V_3 ∶ b≥0\}⊂V_3$. This is called the dual of f under the identity isomorphism of $R^3⟼V_3$.
Short answer: no. The adjective "dual" has precise definitions in several different mathematical contexts. The example you ask about is not one of them.
In your example the fact that the domain has a vector space structure is irrelevant because the function is not linear where it is defined.