AI's favorite/shunned governments:
-5 Favorite government (only applies if both you and the AI are in this government)
+4 Shunned government (only applies if you are in this government and the AI is not)
The worst that can happen is you switch from the AI's favorite government, that you both are in, then only you switch to the AI's shunned government (net result +9). An example would be if you switch from despotism to republic and Egypt, Zulu, or Celts stays in despotism or goes to Monarchy. Or you go to Monarchy and the Aztecs stay in Despotism or goes to Republic. Mongols is the other civ that favors despotism, but democracy is their shunned government.
Despotism: 5 AI
Republic: 6 AI
Monarchy: 6 AI
Democracy: 4 AI
Communism: 3 AI
Despotism: 7 AI
Republic: 7 AI
Monarchy: 4 AI
Democracy: 2 AI
Communism: 4 AI
For the most part, the AI is in Republic-Democracy while at peace and monarchy-communism while at war, regardless of what their favorite/shunned government is. So even though there are an even 6 AI for both republic and Monarchy, the Republic would apply more often for favorite government because the AI would also be in that same government. For attitude purposes, republic would only be bad for you if most of the other AI is still in despotism or going to a war-time government because of wars. If you are still in despotism, while the rest of the world has switched governments (more frequently seen on higher levels), you'll see the AI attitudes worsen.