It is true that in a more modern perspective only gauge invariant operators can be observables, however even in the more traditional way of thinking about gauge theory (say via gauge fixing), the appropriate definition of charges is those which are eigenvalues of the generators of global gauge transformations, in this sense $SU(3)$ quarks have eight (nonabelian) charges, and you can only specify two of them at a time because the Cartan subalgebra has dimension two. Global gauge transformations can still change charges but this is not strange, since global transformations are physical.

On the other hand, the three colors do nothing more than labeling the entries of the $3\times 1$ column vectors, and can't be interpreted as charges.