Citizenship is a different thing and I don't see how it's related to this discussion.
The United States really is one country, and is Constitutionally required to maintain unity, to have only one front for international affairs, and to allow U.S. citizens to be citizens of the state in which they reside.
None of these is true in the EU. EU member states are permitted to leave. Each EU member state carries on its own foreign relations. Each EU citizen maintains citizenship in the EU member state of his or her birth.
The EU is not at all analogous to the United States after the Constitution of 1798 and post-Texas v. White. There might be an argument that it's similar to the United States under the Articles of Confederation. Any insistance that they are more similar is a false equivalency. That is the point of GP.
In the interest of promoting honest discourse, please do not truncate people's statements. If a Californian "fully" moves to Texas, i.e., establishes domicile there, then they become a citizen of Texas. That is a key difference.
As a country, you can move almost anywhere in the US and have full citizenship rights. (Note that Puerto Rico, American Samoa, etc are not part of the US, and you don't get automatic rights when you move there. American Somoans are not even US citizens by birth, for example, because it's not part of the US. Palmyra Atoll and DC are the only part of the US not part of a state, and residents are not citizens of a state and don't have full US citizenship rights.)