As the question title suggests: as far as I know, the Articles of Confederation allows each state to make trade agreements, a right that is not available to today’s EU members. But the United States was officially a united country, while the European Union is not. So how would you compare them on their integrity?
The European Customs Union was a separate agreement. Here just the countries agreed to join their forces to improve their bargaining power with the rest of the world. But even though all EU countries signed it, it doesn't have a lot to do with the political union. The political union is far looser than what you have the USA.
Actually there are a lot of separate agreements binding the countries, Schengen for the border controls, Maastricht for the monetary cooperation and so on, but overall they are not as integrated as the USA. There are no federal taxes, there is an office to coordinate police cooperation, but there is no federal police. Defense is fully in the hands of state governments. Most of the EU agencies have a counselling/coordination role, but no power over the single states.
The European Union and the Articles of Confederation can not be directly compared, because they have very different focus.
The Confederation was mostly about diplomatic and military cooperation. It mandated that the participating states would form a defensive pact ("assist each other, against all force offered to, or attacks made upon them, or any of them") and also coordinate their military actions (States were no longer allowed to wage war without permission of Congress). It did very little to coordinate economic cooperation. Congress had no authority to regulate interstate commerce and states were still issuing their own currencies (there was a Continental dollar issued by the US Congress, but it was plagued by severe stability issues).
The European Union, on the other hand, is mostly about economic cooperation. The core feature of the EU is the European Single Market. The primary goal of the EU is to reduce trade barriers between EU members. So member states had to give up a lot of their autonomy regarding their economic policy. Most EU member states abandoned their own currencies in favor of the Euro, the EU determines tariffs, and the member states handed a lot of control over regulations for goods and services over to the EU government.
The EU also tries to coordinate defense in form of the Common Security and Defense Policy. But this policy is a lot less strict than that of the Confederation. While the CSDP enables the EU to engage in military actions (like the European Union Training Mission in Mali), individual EU member states still maintain their own armies and still have permission to use them to engage in military actions on their own accord (when some EU members want to bomb ISIS, the EU does not stop them). The CSDP does contain a collective self-defense clause, but considering that most EU members (all except Sweden, Finland, Ireland and Austria) are also NATO members, this is rather reduntant.
So which one was the more integrated one? One could argue that from their relative historic perspectives, both were. In the 18th century, military cooperation was far more important. In the 21st century, economic cooperation is far more important.
Under the Articles of Confederation, Congress lacked the authority to regulate commerce, making it unable to protect or standardize trade between foreign nations and the various states. In 1784, Congress requested that the states grant it limited power over commerce for a period of fifteen years, but many of the states did not comply . . . On February 16, 1785, [a] committee recommended amending the articles of Confederation so that Congress would have power over commerce.
So even enforcing the rules of a Common/Single Market would have been impossible under the Articles of Confederation.
Actually, if this paper is not incorrect, the Articles had somewhat weird/narrow stipulations with respect to some foreign partners:
The Articles of Confederation reserved to the states the right to levy duty except such as might interfere with stipulations in the treaties with France and Spain.
A quick search confirms that to be correct (Article VI para 2):
No state shall lay any imposts or duties, which may interfere with any stipulations in treaties, entered into by the united states in congress assembled, with any king, prince or state, in pursuance of any treaties already proposed by congress, to the courts of France and Spain.
It seems to me that a priori elevating some pre-specified foreign countries above others in your constitution is good reason to go back to the drawing board...