Very generous to allow targeting of the string and not the whole bow. I would not allow it. I don't consider the string to be a separate object while it's on the bow. And, given that the string was targeted, I don't see how the weight of the bow had relevance. Similarly, I would not allow targeting shatter on a sword hilt, or armor straps, or a section of floor, etc. Of course, what really is an "object" is a gray area, so I can't fault the DM in this case. But, as a player I would work towards exploiting the precedence it sets up.