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.
Citizenship is a different thing and I don't see how it's related to this discussion.