No, the European Union has not historically and presently aspired to become a superstate, because it has not been set up with enough power to turn into one. The European Union functions more like the United Nations for Europe than like the United States. In Europe, there is so much diversity that becoming a super power is unlikely.
No, the EU has not historically and presently aspired to become a superstate, because their actions do not demonstrate their intent to become a superstate. Consider that the EU has twenty-four different official, working languages. If the EU wanted to become a superstate, they would nationalized language in order to assimilate their countries. By allowing each country to maintain their individual language the EU has demonstrated that they do not intend to become a superstate. The EU concerns itself more with economic affairs than with power and influence.