I did some investigation into this, loading a couple saves in the Modern Era, using HotSeat to peek at what the AI was doing. Caveat #1: this is pre-February update, before Monarchy became more flexible. [I want to finish this game before loading the update.]. Caveat #2: [GS] game is Deity-level on a standard-size map, so many of the AI have a lot of cities (and they produce twice as much as I do).
Of 7 AIs, 3 had not learned Divine Right (for Monarchy), despite knowing at least Nationalism. One even hadn't learned Theology, despite having a Religion. No AI had learned Exploration (for Merchant Republic) or Reformed Church (for Theocracy). (Although, none of these AIs had a religious bias, even though they were engaged in theological conflict in my territory.)
Of the four AIs that could have chosen Monarchy:
two were in Republic, apparently for the amenities trying to keep all their cities content, also employing Liberalism or Civil Prestige or both
one was in Monarchy
one was in Autocracy, but was down to two cities.
Of the three AIs that could not choose Monarchy:
one was in Oligarchy (and conquered someone else's capitol, and was the target of a military emergency)
one was in Autocracy, in a Dark Age, and losing cities to my flipping
one was in Republic, and was getting +10% growth & production bonus for being happy.
The AI seems to care more about the side benefits of a government, than having extra cards. Autocracy for the production bonus. Republic for amenities. Oligarchy for war. They don't need the housing benefit of walls from Monarchy, and most prefer to conquer city-states than suzerain them.
