The republican government clause does not mean that the states and the federal government share power .
That would be the definiton of federalism.
The republican government clause, or more precisly, Clause 1 (entitled "Republican government") of Section 4 of Article IV of the U.S. Constitution, declares that all states should share the principles of a republican government.
It has nothing to do with the Republican Party we know today, but is a political philosophy which is attributed to the Founding Fathers.
Such principles include the consent to be governed: the idea that the government is only legitimate as long as the citizens agree to be ruled by it (unlike inherited monarchy, for example).