(Sorry for this (long) German lesson in the Dutch course. I like etymology...)
(tl;dr: German "Burg" = castle, "Bürger" = originally a special citizen of a medieval city, both are related, but not within the same context or age.)
A "Burg" in today's German is a castle. The kind on hilltops, with thick walls and/or ramparts; they used to be safe spaces (including a well and a place for cattle to graze) for the inhabitants of the village to withdraw to when enemies came around. (Originally, a "burgus" was just a small fortified watchtower, a "castrum" in Latin; cf. Ancient Greek "πύργος / pýrgos" = tower. Then it became the fortified space on a hill, and only later a castle.)
Towns ending in "-burg" were associated with (located below) a "fortified hilltop" or actual castle (or Roman fort). There may, however, have been an influence from Italian "borgo" = "city district; new district outside the city walls; town with commerce and agriculture that's not a città nor a villagio".
Apparently both "der Bürger" (the citizen) and "die Burg" (the castle) are derived from "burga" = "protection", and also words like the verb "bergen" (to salvage e.g. a person from under an avalanche or when they broke a leg hiking on a mountain; or a wrecked ship in a storm / from the bottom of the sea; originally: to withdraw up the mountain - "der Berg" - for safety) and "geborgen" (safe and comfortable e.g. in somebody's arms). It's also related with English "borough".
When cities became a thing in the middle ages, not every inhabitant was a Bürger, but only those with a certain status, rich people who were eligible to be voted into the city council; you had to apply for being registered as a Bürger, and it came with certain rights and duties.
In various parts of Britain (and then eventually in the US), "burg" to mean a fortified place became burgh, borough, brough, and bury. You can look around the island and see which shifts in pronunciation took hold where. Edinburgh is in the north, Middlesbrough a little south of there, Glastonbury and Salisbury to the southwest, etc.
And there's a pretty good correlation between where the Romans set up camp (-chesters and -casters (compare "castle")) versus where the early Saxons or Danes (and other Germanic peoples) were (the -burgs and their ilk).
I accidentally put mayor (burgemeester; whoops). Duolingo informs me that the correct answer instead of mayor was "Am I a hamburger of Belgium?" Um, okay. If we can be bananas, I don't see how this isn't so far out of the realm of possibility. Thank you for the chuckle, Duo. =P
They actually don't. From would be place of origin, like "I am from Albuquerque, New Mexico." That is where you grew up or lived for a long period of time. Of is a specific descriptive for WHERE you are a citizen in this sentence. Your translation would mean that you are literally anywhere in the world right now OTHER than Belgium, informing someone of your citizenship there. The latter would be able to be used in Belgium. As well, you don't see "I am a citizen from..." that often, as it's not very common. From is origin. Of is possessive.