I cannot speak about colour charge, but electric charge sure has a rather interesting geometric interpretation in string theory, in fact, two such interpretations that are equivalent through T-duality.

The first such interpretation is not unique to string theory, and is far from being stringy in nature. It's also present in Kaluza-Klein theory, and also supergravity. Namely, the electric charge is proportional to the momentum of the particle in one of the compactified, periodic dimensions. In other words, the observed electric charge is the number of times the particle oscillates, or "circulates" in this compactified dimension. This is also where the *U*(1) symmetry of electromagnetism arises. In heterotic string theory, this symmetry is unified into the larger \(\operatorname{Spin}(32)/\mathbb{Z}_2\) or \(E_8\times E_8 \) symmetry, which is a superset of \(U(1)\) symmetry.

The other interpretation is much more stringy in nature. That is, the electric charge is equivalent to, in a suitable choice of natural units, the winding number of the closed string. The sign of the charge is given by the direction of winding; this implies that only theories with closed oriented strings allow for this interpretation.

These two descriptions are T-dual to each other, in other words, they are equivalent descriptions, through T-duality.