I think talking about your world's governments is not crucial to worldbuilding for FRPGing. Most of the world, for most of human history, has got by without governments in the contemporary sense of that concept.
Furthermore, governments figure very little in classic fantasy fiction.
What are the governments in JRRT's Middle Earth? No idea. Does the The Shire have any public officials other than a Mayor and some Shirriffs? We're not told. It's modelled on England, but does it have JPs? Are Shirriffs, among other things, analogous to JPs? We're not told. Does The Shire have a system of criminal justice? We're not told.
Gondor is an important kingdom. How is it administered? Who rules Osgiliath, when it is not being sacked by Sauron's forces? We don't know.
What is the government of Bree? Are tariffs levied on imported goods? We don't know.
Or consider Earthsea. This world provides an engaging backdrop for multiple, highly-regarded fantasy stories. How are the lands of Earthsea governed? How are taxes collected, and who commissions public works? Almost nothing about these things is said.
In REH's Hyborian Age, what are the duties of the King of Aquilonia, other than to lead the military defence of the kingdom? We don't know.
My advice to a new GM would be to identify/invent a couple of prominent local officials, and to make a few decisions about them, having some regard to what sorts of themes or adventures they might connect to: they're a wealthy land-owner whose power is owed to that fact (much of mediaeval Europe); they're an elected official whose power is due to their ability to command popular support (classical Athens); they are an urban oligarch whose tremendous wealth grants them tremendous informal power, which then demands that they be granted some official position of authority (a Roman Senator or early modern Italian oligarch); etc.
If that seems like too much work, and too much worrying about political sociology, then my follow-up advice would be to do less work, not more! Grab a title from Gygax's handy list in his DMG (or whatever contemporary version of that is available) and just stick it onto an important NPC.