Why does the pion live in a representation of isospin $\rm SU(2)$ and is the mediator of the strong force generated by color $\rm SU(3)$?
I somehow find strange that this is the case. Given that $\rm SU(2)$ and $\rm SU(3)$ are different groups with different representations, I find it very strange that a particle living in one group is so important for the force created by another group.
Why the pion lives in a representation of isospin SU(2)
Sure, it is well described as a part of the meson octet. That is how the quarks were discovered. Do not forget that SU(2) is a subgroup of an SU(3), in this case weak SU(3).
and is the mediator of the strong force
It is a mediator in some of the models of the strong nuclear force
generated by color SU(3)
The strong nuclear force is not generated, it emerges from the spill over of the color SU(3), similar to the spill over electric forces that are part of the explanation of the binding of the atoms and molecules into solids.
Given that color is confined, any "long-lived" resonance has to be an $SU(3)$ singlet. Pions are no exception. A pion is effectively a condensation of a left-handed quark and a right-handed antiquark. The color $SU(3)$ is non-chiral, meaning left-handed and right-handed fermions carry the same $SU(3)$ charge, thus the $SU(3)$ charges net out in a quark-antiquark pair. That being said, pions are the bound state of the strong force, thus you can deduce some color-related information by whacking them around.
On the other hand, the electroweak interaction is not confining, thus you could have $SU(2)$ singlets, doublets, or what have ya. Given that left-handed quarks carry weak $SU(2)$ charge and right-hand quarks don't, a pion as a left- and right-handed pair can NOT be an $SU(2)$ singlet.
