“So by your argument when Obama claims that the Senate is in recess to exercise a recess appointment, that means that they are not both part of the government. After all they have different perogatives, right?”
Your analogy completely fails. The Senate is the Senate of the United States. The President is the President of the United States. They are part of two branches of government - which has three branches in all. You do know that, right? There’s one government of the U.S. The Senate (and the House) represent one branch - the legislative. The President represents another - the executive. There is, however, only one government despite its three branches.
The Church and the English monarchy were not of the same government, nor were they branches of the same government. Thus, your analogy completely fails.
“Or medieval church and state shared government and authority over the people ,”
No. They did not share authority. They both had their own authorities in their own spheres.
“and disputed over what share each should have.”
There were disagreements, but it was universally recognized that there were two separate spheres.
“They were different faces of the same coin.”
Nope.
Except they shared authority in the appointment of personnel, in collection of taxes, and in the courts.