One way to guess about what should be canon is to look at the creators of the game and setting. Marc Miller mustered out of the US Army, and went to Illinois State University in 1972, where he met by Rich Banner, Frank Chadwick, Loren Wiseman, John Harshman. They created GDW, and all but Banner were credited with creating Traveller. The Keith brothers added a lot to the setting too, but weren't among the creators. All were war gamers and history buffs, particularly interested in World War II and the Roman Empire, and very knowledgeable about US history from growing up in the US and paying attention, and about the British Empire because it was so important in so much military history.
From all that, one can reasonably guess that the canon Traveller setting is a pastiche of the Roman Empire, the British Empire, and the US military. Knowing lots about each -- and about 1970s US university culture -- is a good way to guess what their vision is likely to say.
The Sylean Federation, its transition to the Third Imperium, the barracks emperor era, and a lot about the Third Imperium, can all be seen as the Roman Empire in Space. The structure of the Imperium is most closely modeled on the British Empire before telegraph or radio -- probably with a touch of 1970s anti-colonialism as a counter-influence. The hostile relationship with the Zhodani Consulate is inspired by the Cold War, though it developed into something with more science fiction roots, as a psionic (wannabe) utopian culture. The Long Night is most inspired by the 1970s image of the Dark Ages after the collapse of the Western Roman Empire, with a big portion of Great Depression. The Rule of Man is a blend of the European conquest of the Americas and US "Manifest Destiny". The Vilani imperium is pretty thinly described except is the fragile Goliath waiting for the Terrans to knock it down; the Terrans' initial survival seems analogous to North Vietnam's survival of war with the US and eventual defeat of the US-backed South, while the Terrans' defeat of the Vilani looks like the US defeat of British colonialism and subsequent eclipse of its former colonial ruler. The Ancients are a science fiction trope -- notably explored in the 1968 book and film 2001 A Space Odyssey -- necessary to populate known space with an assortment of humans.
So what does this mean in terms of the governance of the Third Imperium? I think the most direct model is the British Empire.
The Emperor is analogous to the king or queen of England. He or she is in some ways the absolute ruler of the Imperium, but the Imperium is too complex for any individual to rule by decree, so much of the administration is delegated to the Moot, and anything requesting quick response is delegated to nobles who are closer to the scene. The Imperium (unlike the British Empire) was founded to facilitate business, so business interests (both the successors to the Vilani bureaux and conventional corporations) are highly influential (like some of the British Empire's corporations, including the East India Company). Local government is largely beneath the interests of the Imperium, and is mostly left to the locals, as long as they don't mess with inter-system trade, don't use weapons of mass destruction,* and refrain from enslavement of sophonts.
Bringing that to a practical level, an Imperial noble on a world is the local representative of all power that the Imperium claims in a system. His, her, or its word is law over the Starport Authority and other Imperial agencies and facilities, military, civilian, and in between, unless other nobles or non-noble officials are granted autonomy by a higher ranking noble. (For example, if a planetary baron is perceived to be a skilled military leader but a klutz at administration of civilian operations, a civil servant might run the starport under the delegated authority of a count in another system.) On a fief, a noble would be the absolute ruler, unless a higher noble pulled rank.
A noble might enforce his rule over the fief through huscarles, police, security workers, or even trust in citizens of the fief. He or she might delegate some law enforcement to police invited from a planetary government's police force. He might even walk around the fief and shoot people who displease him.
Outside the fief, a noble likely has a status similar to modern diplomatic immunity. He, she, or it is not subject to local laws unless a higher noble has granted the local government powers. ("Baron Eneri is subject to all local laws in effect as of 1105, but exempt from arrest, imprisonment, or mandatory personal appearance in court proceedings, but is subject to fines for local crimes, and fines in lieu of imprisonment according to Schedule 989 Sections 22 through 26. All enforcement actions affecting the baron or citizens of the barony must be reported to the designated representative of the Imperial County within 14 Standard Days.")
A noble may have close relations with a local government, may be unpopular enough to be at constant risk of assassination unless protected by a few grav tanks, or somewhere in between. Nobles might be close to some governments on balkanized worlds and at odds with others.
Individuals may have various types of citizenship: national on balkanized worlds, planetary or system-wide, citizens of a fief, Imperial citizenship, and may hold several of those. A person's citizenship may affect the application of law, or not. As I see it, the only default privilege of Imperial citizenship is ability to pass through Imperial starports; applying for one is a necessary part of traveling outside one's home system. But many worlds may grant limited privileges to holders of Imperial passports, such as exemption from certain laws that are meant for locals, such as mandatory membership in the state church, deference to the planetary Bureau of Supervisors, etc. On other worlds, an Imperial passport that doesn't mark one as also a citizen of the world, one might be subject to tourist taxes, exclusion from citizen-only facilities, and routine bureaucratic stink-eyes.
In short, it varies a lot between worlds.
* Nuclear weapons are considered weapons of mass destruction if used in an atmosphere (or presumably against civilian space habitats), but they're permissible in space combat.