Answer

First of all, you are assuming that whitening teeth on the Sabbath is permitted, and I do not know where you got that from.
Even if it is permitted, distinctions like this exist in many categories of prohibited labor. If a labor is usually done (or was done in the Tabernacle) on a certain type of object (for example, things that grow from the ground), then it is defined as a Torah-level prohibition on the Sabbath only with respect to those things.
But you are speaking from the perspective of the prohibition of whitening. It is possible that this is prohibited because of dyeing (there is a substance that whitens). However, that would depend on whether there is actually a dyeing substance here, or whether the substance simply cleans and the tooth becomes white on its own. In that case, it seems to me that it is permitted.