While Britain and America tend to form a united front in our foreign policy, our constitutions and values are extremely different, and claiming they are the same is inappropriate.
The Special Relationship is presented by dubious minds at the forefront of British foreign policy, like Liz Truss, as if it represents some form of ideological hegemony based on values of freedom and democracy emanating from Washington. In reality, there is no such thing. We speak the same language, but the British state is actually more different from the United States than even the Russian state is.
The last time anyone checked, the United Tsardom of Russia did not still seem to have a tsar and a council of unelected barons who could impede their parliament's will. That, however, is what the United Kingdom still has.
The bureaucracy and the plutocracy
The idea that Britain has anything like the model of democracy in America is absurd. Countless things are considered acceptable in the US that are considered unacceptable in the UK, and vice versa. Take, for example, campaign funding and lobbying. The US has virtually no rules on that whatsoever, considering money to be speech and therefore any restriction on it to be a violation of freedom of speech, whereas the UK has rules. This makes the character of the two regimes completely different, to such a point that they could justifiably deny that each other are true democracies at all. If we were in a mood to quarrel, America would accuse the UK of stifling political opposition with bureaucratic red tape, while the UK would accuse America of being a plutocracy.
Another example is the freedom of speech itself. In the US, that is actually enshrined in the US Constitution as a protection for all speech, no matter how offensive - something Prince Harry referred to as "bonkers", to the consternation of Americans. In the UK, you have the freedom of conscience, but how you disrupt the lives of others with that freedom is very much restrained by the law. The government can't suppress someone's beliefs, but a citizen also can't just go out and offend people, as there are laws against it in Britain.
Then there is the right to bear arms. From a British perspective, this is an absolutely unacceptable, bizarre, and menacing idea. In most cases, even British police don't carry arms, while American police have tanks.
Acting on values would make the UK bash America
The point is, in an actual value-driven world, there are more than enough areas of violent disagreement on ideology and constitution for the UK and US to be sworn enemies, certainly not allies. The facts are enough to ridicule the notion that our coalition is somehow standing up for democratic values. Our own regimes and attitudes to governance are fundamentally different and incompatible with each other, to such an extent that we could completely deny each other's legitimacy as states and yet still be fully compliant with our values.
If the UK were to be a foreign policy adversary of the US, the latter would level a host of meaningful criticisms aimed chiefly at the monarchy and the House of Lords, decrying them as undemocratic features. Stock criticisms of the other regime are standard practice when the US has an opponent. Indeed, the UK maintains highly undemocratic vestiges, doing so for the sake of stability and tradition.
The 51st State
There have been reforms in the UK that appear to be aimed at making Britain more like America, such as the creation of a Supreme Court and calls for the abolition of the House of Lords. Whether the creation of the Supreme Court was for the best is not entirely clear, and there are fairly good arguments from both sides. However, British use of American nomenclature is arguably just superficial, aiming more to help maintain the illusion of a shared democratic culture than make it a reality.
On the other hand, considering how stable and reliable the UK's system had been for centuries, reforming is always a bad call. Healthy forms of conservatism rest on the assumption that there are certainly unknowns in revising anything, and that a stable and decent past provides sufficient grounds not to change things very much, even if we have no evidence that reform will go awry. A system that averts a brutish and short life is a good one, even if it is less democratic.
Most would agree that the UK can learn some good features from the US, and that the US can learn some good features from the UK, but it seems indisputable that the UK has a better system. The UK is not plagued by ideology, sectarianism and sedition, and does not have an overpowered executive branch.
UK system is just better
Countries that followed the Westminster system effectively maintained order even in some of the most historically conflict-prone regions of the world, while those that instead emulated the United States were prone to disorder. What the Special Relationship is not, is anything to do with actual constitutions or political values. The UK is quite different from the US as a regime, aside from the shared language.
The Special Relationship has only ever been a mutual attempt by the US and UK to manipulate the other and use the other as an instrument of one's own interests. The US sees the UK as an aircraft carrier off the shores of Europe, while the UK still sees the US as a youthful successor to the British Empire.