That is equivalent to any one in any state in the US can move to another state at will. It just seems odd because they are such distinctly different cultures that have give them their unique identity. I would imagine that someone outside the states would be hard put to tell expalin the difference in the culture of people of North Dakota and Nebraska. With few exceptions, states don't have a big cultural identity.
It's not equivalent because your immigration policies are not set by individual states, but are set by your government. The EU does not have a government with the power to set global immigration policies.