Doesn't "Ich bin Buddhas Schüler" imply that Buddha only has one scholar?
Not necessarily, but it isn't very clear about that either. Ich bin einer der Schüler Buddhas (one of the scholars) or Ich bin ein Schüler Buddhas (another translation for the original sentence) might be less ambiguous.
Why is "Ich bin ein Student Buddhas" not correct? I was taught that Schüler was mostly used for schoolchildren.