If I have a dual citizenship in Sweden am I a citizen of the EU?

Yep, that is what it means HOWEVER its tied to more then just citizenship. Its also tied to WHERE you live. If you were a Swedish citizen, which you are, living in Sweden, then you would get the EU rates most likely. Its that way because that is where you pay taxes and pay into the system that helps fund the schools. Without helping to fund them somehow you don't normally get any preferred rate.

You are an EU citizen. The college tuition would depend on the policy of the individual college. Some don't consider citizenship as much as they consider place of residence.

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