In the U.S., colleges are institutions that award undergraduate degrees: associates degrees (2-year) and bachelors degrees (4-year). Some colleges are stand-alone institutions, such as Amherst College and Williams College (two very prestigious schools). Others are part of a university. For example, Harvard College is the undergraduate division of Harvard University. Another example: the University of Notre Dame includes a College of Arts and Letters, a College of Science, a College of Business, etc., and those colleges award undergraduate degrees. Graduate students are enrolled in the university (which is the same place and often the same buildings and classrooms).
So even if someone begins their higher education at a University, they will first be enrolled in one of the university's colleges... and since most students are undergraduates, they are correct in saying "I'm a college student."