renormalize said:
The statement you referenced is simply saying that the eight ##3\times 3##-matrix generators of ##SU(3)_{\text{color}}## are conveniently realized as the set of Gell-Mann matrices (GMm), but that there are also many other possible matrix-representations that are equivalent, i.e., that satisfy the same group commutation relations as the GMm. The physics of QCD is invariant under the choice of any particular realization of ##SU(3)_{\text{color}}##, so the GMm are chosen for convenience. Note that this is directly analogous to the three ##2\times 2## Pauli-matrix generators of the 3D rotation group ##SU(2)##. Other realizations are possible, but the Pauli matrices are taken as the standard choice.
So the answer to your question is no, there is no other group that can replace ##SU(3)_{\text{color}}## that is consistent with the plethora of experimental evidence for QCD gluons and quarks, just like no group can replace ##SU(2)## to describe rotations in 3D space.
In a gauge theory, the only thing that's fixed by the choice of the group is that the gauge fields transform in a specific way such that covariant derivatives, ##\mathcal{D}_{\mu}=\partial_{\mu} + \mathrm{i} g A_a^{\mu} \hat{T}^a##, applied to a field that transforms under the gauge group under a given representation of this group, the ##\hat{T}^a## are a basis of the corresponding representation of the group's Lie algebra. The gauge connection, $$F_{\mu \nu}=\frac{1}{\mathrm{i} g} [\mathcal{D}_{\nu},\mathcal{D}_{\nu}]$$ then transforms necessarily with the adjoint representation of the gauge group.
You are still free to choose any representations for the "matter fields", and which representations are useful (and also which gauge group is useful) for a specific set of physical phenomena, must be deduced from observations. The Standard Model is the result of a fascinating interplay between theory building and experiments, which themselves were often also inspired by predictions of the theory.
As it turned out, in the case of the strong interaction, the gauge group SU(3) ("color gauge group") was the right choice. This was inspired by the fact that in the old flavor model by Gell-Mann and Zweig for the Hadrons, in order to allow for a particle like the ##\Omega## baryon, which was predicted by this model as being a bound state of three strange quarks and part of a specific representation of another SU(3) group ("flavor SU(3)"), which was introduced as a global symmetry to bring order into the growing zoo of hadrons, which was very successful (and an extension of the corresponding SU(2) version, invented as the "isospin formalism" by Heisenberg with the proton and neutron transforming as a doublet under this SU(2) symmetry group). The problem with the ##\Omega## was that it couldn't be built from three quarks, which necessarily had to be fermions to give the right phenomenology for the baryons (fermions) and mesons (bosons) described as bound states of three quarks or of a quark and an antiquark, respectively. The solution was to assume that each type ("flavor") of quark (then only up, down, and strange) comes in three colors as an additional intrinsic degree of freedom. Then one could build all the hadrons in the proper way with fermionic quarks and antiquarks, if each quarks come in precisely three colors, and that means that the color symmetry SU(3) must be realized for quarks as the fundamental representation (called simply 3), i.e., with a three-dimensional SU(3)-color spinor field. The anti-quarks then necessarily must transform with the conjugate complex representation, ##\bar{3}##, which is of course also three-dimensional but not isomorphic to the 3-representation.
Another rule was that observable particles are color neutral, and with this rule all the hadrons could be built as color singlets and various representations of the flavor SU(3).
Finally it turned out that the strong interaction can be described by describing the strong interaction between quarks by making the color-charge symmetry local, i.e., use it as the gauge group of quantum chromodynamics. It also turned out that such non-Abelian gauge theories are asymptotic free, i.e., that the running coupling constant (in the sense of the renormalization group paradigm applied to this QFT) becomes small in the UV, i.e., at large collision energies. This gave a hint that these theories are also "confining", i.e., that no colored fields can be asymptotically free and thus that within such a model there are no observable color-charged particles, and indeed nobody has ever seen a free quark or gluon but only colorless hadrons. Also with help of lattice-QCD calculations, it has been demonstrated that QCD indeed leads to the correct quantitative description of the hadrons as color-neutral bound states of quarks and gluons, including the correct prediction of the hadronic mass spectrum.
