Please cite the principle of international law which requires the present day iteration of a state's government to have had past administration of a breakaway territory in order to assert a claim of ownership over said territory.

Please also cite any supporting state practice and opinio juris.

How do you go from there - economic dependence and decreasing recognition - to not being self sovereign? They run a government and have elections. As another hexbear pointed out

At a basic level, to be a sovereign country is for the people of that country to have the ability to determine their own collective destiny. Now, sovereignty is not a simple binary but a scale since powerful countries have the potential for greater influence than smaller countries who must fight against the influence of larger countries.

Vietnam has sovereignty. It has an independent military that is battle-tested through winning numerous wars against its neighbors and the US, it has a seat within the UN where it can lobby its interest before a global body of nations, it has international treaties with numerous countries and is free to sign more or back away from treaties if it's in its geopolitical interests, it is part of many international organizations like ASEAN, and it has an extremely savvy ruling party who knows how to play off the blocs against each other for Vietnam's benefit. It's even taking steps to be completely food independent so they won't get fucked over by sanctions and climate change. The only real mark against their sovereignty is the PRC (and ROC) presence in the SCS.

Taiwan, in contrast, has little to no sovereignty. Its military is completely dependent on the US. If it wasn't for the 7th Fleet constantly bailing out Taiwan, Taiwan would've long since been reunited with the Mainland. It has no seat in the UN. A grand total of 12 UN states, many of them Pacific islands that Taiwan constantly bribes for their continued recognition, plus Vatican City recognizes Taiwan. Because Taiwan is not a UN state, it cannot belong to a lot of organizations. Just a few days ago, Taiwan got expelled from the Central American Parliament. The Central American Parliament isn't some hugely important organization and that's part of the point. Taiwan has already been shut out of important organizations like the UN and the WHO and now they're even being shut out of even less important ones. Taiwan has to compete in the Olympics under the humiliating title "Chinese Taipei" and instead of boycotting the Olympics, they choose to compete with that humiliating title, further cementing their inability to move beyond what the PRC and the rest of the world has placed them in. Neither the KMT and nor the DPP are pursuing policies that would bolster Taiwan's little sovereignty, with the KMT thinking if they can kiss the PRC ass enough times, the PRC won't invade Taiwan and with the DPP thinking if they can lick Uncle Sam's boots enough times, the US would save Taiwan and not abandon them like the US did with Afghanistan. Taiwan is also overly dependent on trade with the PRC and in general, Taiwan's economy is intertwined with the PRC, meaning if the PRC does shit like temporary ban some Taiwanese imports, the entire economy feels the strain.

This is a country that's economically dependent on one country and militarily dependent on another country. This is not a sovereign country. This is a pawn that's being played by two countries that are belligerent with each other.

The dispute at this point is over how we define a country, especially because Taiwan clearly falls in a grey area within that definition. I claim that they are fundamentally unable to exercise their sovereignty given they aren't formally recognized as a country by even their greatest allies and benefactors, thus they fail. You claim that they can fulfill the roles of the state, have a national identity, and have various semantic work-arounds for that fundamental illegitimacy, thus they pass. There's also the question of the legitimacy of their founding, with me saying that the ROC was originally an oppressive colonial military dictatorship, but then you would say that it's been long enough and their government has changed enough that it doesn't matter, then we bicker over what constitutes a democracy.

Ultimately the argument would continue indefinitely and I don't think there's much chance either of us would be convinced by the other.

As an aside, the point of the prior comment was that surveys of beliefs can very easily be detached from reality, and so aren't good evidence for claims.

Except no country or international institution would agree with your criteria for a nation-state since that definition also gives legitimacy and sovereignty to lovely people like ISIS when they administered a huge chunk of Iraq or any number of autonomous or semi-autonomous breakaway regions that the international community consistently refuses to acknowledge as sovereign states.

Yeah! Whether other countries let you have a seat in the UN or not is not relevant to sovereignty.