I think that the fairest thing to say is that the exact nature that the writers have intended for the UFP to have has evolved over time.
When TOS first started, in fact, the writers hadn't invented the Federation yet. The Enterprise
was described as being a "United Earth" starship in "The Corbomite Maneuver;" clearly, United Earth was the state that our heroes originally were envisioned as serving. The Federation and its Starfleet were first established in "Court Martial."
In "Journey to Babel," we hear reference to tensions amongst the Federation's Members over whether or not Coridan will be added -- tension so high that it could apparently lead to war. The people making the decision over whether Coridan will be added are called "Ambassadors" -- suggesting that the Federation is now to be seen as a parallel to the United Nations.
Throughout the course of TOS and the early TOS movies, however, we very clearly see that Starfleet is now regarded as a Federation organization -- setting the Federation apart from the United Nations, in spite of the intended allegory, by giving it an armed force and, thus, a legitimate authority to use violence, one of the primary characteristics of a state. So the allegory of "UFP = UN" is starting to bend here, and now the UFP is starting to resemble an interstellar state. The allegory is made all the more statist, so to speak, when we encounter non-Starfleet individuals who have the legal authority to place McCoy under arrest in Star Trek III
, calling themselves "Federation Security," implying a Federation-level law enforcement organization. We further hear of a "Federation Council," but we hear very little of it.
So, as with United Earth before it, no substancial information about the Federation that we might use to link it to any one particular modern state for allegorical purposes exists... until Star Trek IV: The Voyage Home
. Here, we encounter for the first time an individual identified as the President of the United Federation of Planets. We see the President giving orders to the Federation Starfleet, and we see the Federation Council and President holding Admiral Kirk's court-martial. This starts to suggest the United States, at least insofar as they use both the American concept of a "president" and most of the characters retain American Midwestern accents. That the Federation Council has the authority to stand as jury in a Starfleet court-martial also suggests that the Starfleet exists because the Council has raised it -- in the same way that the armed forces of the United States exist because Congress has raised them. Still, we don't hear about the President or the Council making binding laws on Federation Members, or determining all foreign policy, and Star Trek IV
also suggests that Kirk and Company were able to avoid being apprehended by the Federation simply by virtue of being on Vulcan -- a planet that is supposedly a Federation Member. This seems inconsistent with the idea of the Federation as a state, since, after all, one could not very well avoid capture by US Federal authorities just by hanging out in Massachusets.
In Star Trek VI
, however, the depiction of the Federation takes another step in the direction of "interstellar state," and starts to suggest a pseduo-American allegory. We see the Federation President conducting foreign policy towards the Klingon Empire on behalf of all Federation Member worlds, even negotiating and signing a binding peace treaty. The President is also the target of an assassination plot that the conspirators believe will lead to a war, in parallel with the assassination of the Klingon Chancellor, who is clearly the Klingon head of government. So the implication seems to be that the Federation President is both head of government and head of state.
We also see the President more explicitly being depicted as having complete authority over Starfleet -- proposals are made to the President and not the full Council, further suggesting an American model (since, after all, the US armed forces are under the operation control of the US President and not the US Congress). Still, by this point, there have also been several examples of Starfleet taking orders from the full Council in TNG; this may be seen as contradicting an American model, or, at the very least, as complicating it.
TNG, however, brings us a step further in the direction of "Federation as interstellar state" model when we see that the Federation Council, in "Forces of Nature," has declared a Federation-wide "speed limit" of Warp 5. We have previously seen the Council making decisions that are binding on Starfleet, but, if I recall correctly, this is the first time we see that the Council can make laws that are binding on everyone within the Federation. The legislative nature of the Federation Council is re-enforced with references to the Council debating over whether or not to ratify the Federation-Cardassian Treaty in TNG's "Journey's End;" treaty ratification, in addition to once again establishing the Federation's authority to conduct foreign affairs and making binding law over its Member worlds, treaty ratification is a clear trait of a state's legislature. With these episodes, then, it becomes clear that the Federation Council is a legislature.
DS9 brings us back to a more explicitly American model. In "Homefront"/"Paradise Lost," the Federation Starfleet is once again depicted as answering primarily to the Federation President. When Sisko and Leyton propose an upgrade in Starfleet security and in security on Earth, they propose it to President Jaresh-Inyo, not the full Council. The President is also referred to as being Starfleet's "commander-in-chief," further suggesting an American model. Jaresh-Inyo is also referred to as the "elected President;" this would seem to clearly establish an American model, at least for the Federation Presidency, since the head of government in a parliamentary system is usually appointed from the parliament. (The State of Israel was a brief exception to this rule in the early 21st Century, when the Prime Minister was popularly elected.)
On top of this, the Federation President -- in spite of his not being from Earth -- is clearly depicted as having the authority to place Earth, a Federation Member world, under martial law. This would seem to solidify the Federation-as-interstellar-state model, as opposed to a Federation-as-UN or Federation-as-alliance model; confederations and alliances do not have the authority to place their members under martial law and direct central control, but states do. NATO cannot place France under martial law -- but Great Britain can certainly place England under martial law, and apparently the Federation can place its Members under it, too.
A later DS9 episode, "Extreme Measures," further establishes the existence of a Federation Cabinet. This would seem to be the final nail in the coffin of any view other than that of the Federation as an interstellar state; states have cabinets, alliances do not. This would be compatible with both a parliamentary or American-style presidential system of government, but previous evidence, as I outlined above, indicates an American-influenced model.
One of the primary traits of a state is that all politics references or flows back to the state itself; authority is derived from the state. We've already seen other state-like traits by the time of TNG/DS9 -- it controls a specific territory, it has the monopoly on the legitimate use of violence, it is recognized by other interstellar states such as the Klingon Empire, etc. The state nature of the Federation is further driven home in TNG/DS9 through numerous references to authoritative Federation governmental bodies:
Bureau of Agricultural Affairs
Bureau of Industrialization
Bureau of Planetary Treaties
Central Bureau of Penology
Department of Cartography
Department of Temporal Investigations
Clearly, all relevant bureaucracy is built around the Federation by this point -- another clear trait of a state.
Several episodes of TNG, DS9, and VOY also make reference to a Federation Constitution and a Federation Charter. The Federation Constitution is referenced in "The Drumhead" and "The Perfect Mate" (TNG); in "The Drumhead," it is established that the Constitution contains twelve "Gurantees" ensuring individual rights. The Seventh Guarantee is referenced, and it is clearly based on the United States Constitution's Fifth Amendment protecting suspects from being forced to give self-incriminating testimony. "Author, Author" (VOY) also refers to a Guarantee relating to artists' intellectual property rights. DS9's "Accession" establishes that the Federation Charter bans caste-based discrimination. Clearly, then, the Federation Charter/Constitution is at least in part inspired by the US Constitution.
The Federation is also established to have a judicial system with ultimate power of judicial review in DS9's "Doctor Bashir, I Presume?". In that episode, Julian Bashir's father vows to fight the Federation-wide ban on genetic engineering "all the way to the Federation Supreme Court," further implying a US-style government.
So over time, the depiction of the Federation changed from that of a UN-style organization to that of an American-style interstellar state. However, even in later films and series, the Federation is not strictly
US-based. Why? The Federation Council is consistently depicted as having far more authority over the operations of Starfleet than the US Congress does over the American armed forces, and more influence over foreign affairs. In "Valiant
(DS9)," the Federation Council sends a message to Ferengi Grand Nagus Zek proposing an alliance during the Dominion War; they have the power to serve as jury on court-martials in ST4; they give operational orders to the Enterprise
crew in "The Defector" (TNG); they determine that the Founders will not be given the cure to the morphogenic virus in "The Dogs of War" (DS9). The US Congress, while active with oversight, does not have the kind of operational authority over the US armed forces that the Federation Council does over Starfleet (as current history is no doubt demonstrating). In the US system, there is a clear separation of powers, and operational control of the armed forces falls to the President (though Congress retains oversight rights). The lack of a clear separation of powers between the legislative and executive branches, and the increased involvement of the legislature over Starfleet, suggets parliamentary influences.
So, to me, the implication would seem to be a primarily American-based model, with some parliamentary influences. Clearly, the President is popularly-elected, but, clearly, he must share more power with the Council -- and the Council with him -- than is typical in American presidentialism. It's obviously strongly influenced by the American government, but there's a distinct suggestion of parliamentary influences. Either way, though, the depiction of the Federation has generally remained consistent with the characteristics of constitutional liberal democracy (though, interestingly enough, it has never been established how members of the Federation Council are determined).