OK, maybe a historical view might help.
After WWII, the idea of European unification came up in order to avoid another war that would devastate the continent. Because of its importance for the arms industry, the idea, mostly put forward by French foreign minister Robert Schuman, was to put the coal and steel sector under common administration (the Council of Europe, as said, had already been founded in 1950, but its competences are slim - it’s more of a debating platform). In 1952, the European Coal and Steel Community (ECSC) was founded, with France, Germany, the Netherlands, Belgium, Luxemburg, and Italy as original members.
The Treaties of Rome, which came into force in 1957, founded two more communites: The European Economic Community (EEC) and the European Atomic Community (EAC/Euratom). Each one was based on its own treaty, and the three communites functioned parallelly, but the organs were merged - the Commission was exercizing its powers as Commission of the EEC in one case and as Commission of the EAC in another, depending on which treaty it was applying. The ECSC was a special case because it had a slightly different organizational and terminological framework.
Over the years, the membership was increased from the original six to (now) 25: In 1973, Britain, Denmark and Ireland joined; in 1981, Greece; in 1986, Spain and Portugal; in 1995, Finland, Austria, and Sweden (the Norwegians voted against accession in a referendum). In 2004, ten East and Southern European nations joined. Parallelly, new treaties were concluded between the member states, amending or supplementing existing treaties. Important ones were the Single European Act of 1986 and the European Union Treaty of 1992. In 1992 as well, the European Economic Community, by far the most important one because its treaty covered the process of economic harmonization, was renamed the European Community (EC). In 2002, the ECSC Treaty, which was limited to a period of fifty years, expired (which didn’t make any difference because everything the ECSC Treaty covered can also be based on the other treaties). The whole cmplex is now left with the European Union and the two remaining European communites, namely the European Community and the European Atomic Community (the latter one is of very minor importance).
The problem is that the framework wasn’t installed instantaneosly in one single constitutional document, as you Americans did in 1787.