Is it harmless to count three, say, up quark particles instead of one up quark with three colors, as well as 8 different gluons instead of one with different color?
Color is the strong interaction analog to charge in the electromagnetic force. The term "color" was introduced to label a property of the quarks which allowed apparently identical quarks to reside in the same particle, for example, two "up" quarks in the proton. To allow three particles to coexist and satisfy the Pauli exclusion principle, a property with three values was needed. The idea of three primary colors like red, green, and blue making white light was attractive, and language about "colorless" particles sprang up. It has nothing whatever to do with real color, but provides three distinct quantum states.
The rationale for the concept of color can be highlighted with the case of the omega-minus, a baryon composed of three strange quarks. Since quarks are fermions with spin 1/2, they must obey the Pauli exclusion principle and cannot exist in identical states. So with three strange quarks, the property which distinguishes them must be capable of at least three distinct values
The difference with the electromagnetic charge is that color charge can be exchanged. An electron cannot change its charge and still be an electron, it is identified by its elementary charge. A quark can be either red blue or green with color exchanges with gluons, whereas photons do not carry off charge when interacting with electrons.
Color charge acts more like classical positive and negative charges which can be exchanged in classical interactions. The SU(3)xSU(2)xU(1) standard model successfully describes particle interactions, and there the only differenc in quarks comes from the fixed quantum numbers of strangeness , charm, bottomnes and toppness, which are conserved in strong interactions.
The short answer is no, it's not harmless.
The technical reason why is that experiments confirm that the gauge group of QCD is $SU(3)$, which implies three colors for quarks and eight charges for gluons.
The less technical reason why is that if you look closely at particle collisions, the frequency of certain types of collisions is only accurately predicted in the case that quarks have three colors and gluons have eight charges. (These are known as three and four jet events in electron positron collisions, D. Decamp et al. (ALEPH), Evidence for the triple gluon vertex from measurements of the QCD color factors in Z decay into four jets, Phys. Lett. B284, 151 (1992).)