You are allowed to make a notation on the primary AMCAS application of any other standardized tests that you took, including the scores that you received. The example given on the website is taken directly from the GRE, so someone could take the GRE and give the adcoms the ability to consider their scores on that particular exam.
How the mentioning of such scores affects one's application, however, is impossible to say; I have neither met nor read about anyone disclosing other standardized test scores and revealing how it affected their application (assuming that the turbidity of this process even allows them to make such a determination). Moreover, given that the vast majority go through the process making available only their MCAT scores for adcom consideration, I would never suggest taking additional standardized tests solely to boost one's application. Perhaps there are others here with more experience who can comment on how additional standardized test scores may affect applications.