You're exactly right, drag should be measured in units of force, like Newtons. However, although there are small differences in the gravitational constant across the surface of the Earth, a reasonable rule of thumb is to assume it to be constant. Accordingly, drag force in Newtons is proportional to kilograms by a constant term very close to 9.8 m/sec^2, so 1 N of drag force is approximately equivalent to "100 grams of drag."
That said, I try not to use "grams of drag" and instead use Newtons of force. But, as you've undoubtedly seen, many others (including bike manufacturers and wind tunnel operators) do use "grams" so in order to promote effective communication, I'll switch to using a term I know they'll understand.