As Pope Benedict XVI said in his encyclical Deus Caritas Est, they have distinct spheres, but they are always interrelated. The Church should inform the decisions of the state since the state must act according to the truth (Catholic social teaching is a perfect example of this principle in practice). Likewise, the state has the duty to protect the rights of the Church and make sure she can exercise her mission as the Second Vatican Council and past and present Roman Pontiffs have declared.
Leo XIII went into more detail on why this is saying that the state governs the temporal while the Church the spiritual, but both should work together in harmony since both are subject to the same God and also have the same end–the common good of the people entrusted to their care.
Catholicism would not be properly called a state religion because it is supranational (state religions are things like the Church of England or the Church of Russia during certain periods which were essentially cabinets of the state governing body). However, it can be the established religion of a state. However this is not always prudent as it can be scandalous when the state does not operate under Catholic principles. A good non-Catholic example of this is how the Greek Orthodox Church is the official religion of the state of Greece, yet Greece is one of the most secular and immoral states in Europe.
In my opinion, since there are few if any Catholic states, the current concordats fill our needs without us being unequally yolked.