A graph property is a class of graphs with the property that any two isomorphic graphs either both belong to the class, or both do not belong to it.
A graph invariant is any function $S:G \to {R}^+_{0}$ such that for isomorphic graphs $G,G'$, i.e. $G \sim G'$, holds $S(G)=S(G')$.
If there is now an algorithm being able to tell isomorphic graphs apart, $S(G) \neq S(G') \Leftrightarrow G \not\sim G$, that is between $P$ and $NP$, how can I then interpret that a graph property is in $NP$?
I am somehow lacking here a piece that telling graphs apart is between $P$ and $NP$, and hence any graph property should be bound by such a complexity, and still having properties in $NP$.
Even being instantly able to tell whether two graphs are isomorphic doesn't give you an obvious test for some graph properties.
Consider the property "$G$ has a Hamiltonian cycle": testing for this is one of the classic NP-complete problems. You could try to test for it by testing if $G$ is isomorphic to any graph on a list of all $n$-vertex graphs with a Hamiltonian cycle. However, there are exponentially many graphs on any such list. So you will not get an efficient algorithm in this way.