This answer is informed by two things:
1. The Sword Coast Adventurer's Guide (page 7-20)
2. Human Political Organizational Habits from RL History
The entire Sword Coast is not listed as "The Sword Coast."
Its structure (as any structure is) is a fusion of the feudal period before and after the reign of Charlemagne as emperor of the Holy Roman Empire, the model of the ancient Greek city-state, and an allusion to extranational organizations such as the Hanseatic League of the Middle Ages along Northern Europe's coastal borders , (FWIW, the world of Greyhawk lore has a similar association called The Iron League).
City states with medieval feudal overtones
As the kingdoms further east include Cormyr and further south Amn and Tethyr, the Sword Coast region is characterized by urban junctions separated by long wilderness areas and sparsely populated areas. These urban centers are a strange cultural fusion of medieval, feudal and renaissance cities (led by nobles). They are all forces for themselves. Waterdeep, Neverwinter, Silverymoon, Luskan, Baldur's Gate, Daggerford, Mirabar, Longasaddle, Mithral Hall, Yartar and Amphail are polities of various sizes mentioned in SCAG.
There is no king, there is no emperor. There may have been a greater kingdom or empire, but now there is none. Every city-state has its own government and control mechanisms. (Or try it).
How is the larger area of the Sword Coast managed?
Is not it.
How are laws enforced and by whom?
On site by local authorities
How are officials appointed?
Locally. Each locale has its own method.
How are taxes collected?
You will find mentioned in the Water-deep dragon robbery, the Murder and Baldurstor, and others published adventure notes on taxes that were paid to a local authority.
Imagine the PCs joining some orc robbers and slaughtering a city. Who would come to enforce the law of the country?
Either (1) no one, (2) a local force under the authority of a local lord, or (3) another group of adventurers / mercenaries (led by the DM as NPCs in a given game). But if what the PCs did was outrageous enough, their raid could grab the attention of a faction or supranational organization like the Lords Alliance. Or the Harpers. These organizations could put together the resources to track down the PCs and bring them to justice. (And that's all the DM needs to do in a particular game or by the author in a particular FR novel.)
Trade connects all cities.
As with normal human models of civilization, trade between different regions and city-states leads to common interests and rivalries. Which leads to …
The role of factions as an in-world mechanism for political intrigues
Between the Harpers, the Lords Alliance, the Zhentarim, the Emerald Enclave, and a variety of nongovernmental organizations (see Knights Templar in the Middle Ages / Crusader Era for an excellent parallel) organizations with goals that are not tied to a local area act to their own interests to promote several areas.
- You can see similar "non-state actors" in an organization like that
Masonic fraternal organizations (as well as the conspiracy theory
Versions of it) or a more political organization like Al
Qaida or DAESH: They are not resident in a country, but a common interest or interest
shared view of the world.
The Lord's Alliance: An Alliance Instead of "Governance"
Please see page 7 of the Sword Coast Adventurer's Guide for a description of The Lord's Alliance.
A Confederation Among the Rulers of Various Northern Settlements … In addition to their military support and a forum for peacefully disseminating differences, the Alliance has always acted on the principle that common-sense communities that trade are less likely to disappear with each other to lead. By maintaining strong trade relations inside and outside the Alliance, the Lord's Covenant contributes to the maintenance of peace.
As mentioned above, and consistent with the historical ties that exist between organizations such as the Hanseatic League, trade and "collaborations where there is a common interest" is as far-reaching as a "supranational" organization.