The gluon with color (|g1> + |g2> + |g8>) / √3 is just a superposition of the gluons with colors g1, g2 and g8, the same way you can make superpositions of any other particles. You are right that the choice of basis vectors is arbitrary, but that doesn't make it wrong to count the number of dimensions. It also doesn't make it fundamentally different than, say, polarizations of photons or even flavors of quarks. You can have superpositions of photon polarizations or quark flavors.

All of these are continuous properties in an n-dimensional vector space.