Sure, it's possible. Stickier paints are also softer, so a custom may well have softer paint, and therefore a tightly-wrapped string could theoretically dig in to it. Something to watch out for.
Most production-run tamas probably have paint that is tough enough for this not to be an issue. Although Sunrise, for example, do string some models with a small strip of stiff paper between the string and the tama, to prevent this issue.