Because sentences such as "I am student" or "I am teacher" are ungrammatical in English - an article is missing. By contrast, German normally doesn't use articles when talking about occupations.
PS: I've just seen that your native language seems to be Farsi. Using the articles correctly can be tricky, especially if they don't exist in your native language. So you should be extra careful as to how they are used. :)
If you want to avoid the gendered expression, a simple solution is to say "Ich studiere." In official documents (but only there) the preferred term is the gender-neutral "der/die Studierende". If you want to use the words "Student/Studentin" it sounds awkward if a female student says "Ich bin Student".
Generally, there are no colleges in Germany. The system is different. Students go to "Universitäten" (universities) or to "Fachhochschulen" (universities of applied sciences). But yes, you would use the German word "Student" when talking about e.g. American college or university students.