For a good synopsis of Taiwan's diplomatic situation click here. It's John Oliver's take on the subject.
My two cents? It's not easy to explain Taiwan's status with regard to other countries and international organizations. Explaining this status to people who are new to the topic will also try their patience. In some respects you have to live Taiwan for a while to get it, and expecting people outside Taiwan to understand China's position, Taiwan's position, and the U.S. policy of "strategic ambiguity" is asking a lot.
To begin, I'll start with the Wiktionary definition of "country." It is as follows:
"1. (chiefly British) An area of land, a district, region.
2. A set region of land having particular human occupation or agreed limits, especially inhabited by members of the same race; speakers of the same language etc., or associated with a given person, occupation, species etc.
3. The territory of a nation, especially an independent nation state or formerly independent nation, a political entity asserting ultimate authority over a geographical area, a sovereign state."
Wiktionary contains two other definitions of "country," but they have no bearing on the issue of sovereignty. I'm also setting to one side the Chinese definitions of "country" (or 國家) because I don't think the meanings of this term in Chinese have as much to say about the international context in which Taiwan exists.* In a regional sense the Chinese definitions aren't as important, simply because Taiwan shares the region with several nations in which Chinese is not an official language.
According to Definition 1 Taiwan is definitely a country. But yes, that definition is not used much outside of Britain.
Definition 2 is more problematic. How do we define "agreed limits?" Certainly some groups in Taiwan agree on limits, but the limits agreed upon by the DPP (which would rather see Taiwan as an independent entity) and the KMT (which, historically at least, has regarded China as its rightful territory and Taiwan as a province within that territory) are very different. In a larger sense there's also the question of who "owns" or "controls" what, and whether Taiwan is part of China or China is part of Taiwan.
The terms "race" and "speakers of the same language" also pose problems. Are we talking about ethnically Chinese people? If so, then what about the aboriginal tribes living in Taiwan? And if we're talking about ethnically (Han) Chinese people, are the Han Chinese living in Taiwan really the same as those in China? Is Mandarin spoken the same way? Is the culture the same? How do we define "race" exactly, and who is doing the defining?
Definition 3 fits Taiwan. China does not administer or exercise political control over this island and its associated territories. Unless, of course, you ask most of the politicians here during a press conference. Threats and intimidation from the Mainland count for a lot, and these politicians often try to avoid antagonizing China.
If I look up the Wiktionary definition of "nation" I get this:
"1. A historically constituted, stable community of people, formed on the basis of a common language, territory, economic life, ethnicity and/or psychological make-up manifested in a common culture.
2. A sovereign state."
Taking Definition 1 into consideration, is Taiwan "a historically constituted, stable community of people"? Yes, definitely. Taiwan has its own history, or perhaps more importantly a series of historic events which unite people within a single culture. The "formed on the basis of a common language" is troublesome when you factor in changing attitudes to Taiwanese/Hokkien and aboriginal languages, but yes, I think that people here are bound together by an understanding of Mandarin Chinese. The "psychological make-up" part of this definition seems an easy fit, especially given that most Taiwanese people have never lived in a place that hasn't been threatened by the designs of Mainland China.
Definition 2 works as well, and in case you're wondering, "sovereign" is further defined as "exercising power of rule," and in turn "rule" is defined as "The act of ruling, administration of law, government, empire, authority, control." Yes, Taiwan is a country by this definition.
Of course there's also the Wikipedia entry for Taiwan, which states:
"Taiwan, the Republic of China (ROC), is a country in East Asia...
...the political status of Taiwan is contentious. The ROC no longer represents China as a member of the United Nations, after UN members voted in 1971 to recognize the PRC instead. Meanwhile, the ROC continued to claim to be the legitimate representative of China and its territory, although this has been downplayed since its democratization in the 1990s. Taiwan is claimed by the PRC, which refuses diplomatic relations with countries that recognize the ROC."
PRC, ROC... you've got a headache now, right? So does the rest of the world. Suffice it to say that most people in Taiwan would prefer to be recognized as an independent nation, and to not be reabsorbed into communist China. All people in China (at least if you ask them while the cameras are running) contend that Taiwan is their territory, and that it will eventually be reabsorbed into China one way or another.
I could go into the reasons behind this argument, but that would be a whole other entry's worth of stuff. The list of reasons would furthermore extend back to the Ching (Qing) Dynasty and maybe earlier, and would be compounded by an attempt at colonization, a world war and a lot of human suffering. Let's not go there today.
China's position on Taiwan is easily understood from the above quote, questions as to the legitimacy, duration and scope of various political entities aside. But what about neighboring countries? What do they think of this issue? And what about the United States, without which Taiwan in its present form wouldn't exist?
From The Diplomat, on Japan-Taiwan relations:
"Japan-Taiwan relations - though still unofficial - are shifting quickly in 2021. There are plenty of other examples: a Japan-Taiwan-U.S. trilateral meeting in July, Japan's defense white paper discussion of Taiwan, and several statements in support of Taiwan by Japanese political leaders. Japan's public support for Taiwan reflects a growing consensus in Tokyo that Taiwan's security has a direct impact on Japan..."
In other words, Japan supports the U.S. policy of "strategic ambiguity." It would rather keep China at arm's length, and a Chinese navy with unfettered access to the Pacific (via Taiwan) would pose a real and immediate threat to Japan.
From Taiwan News. Taiwan News isn't the best source of information, but this is the most recent article I could find on Taiwan-South Korea relations:
"South Korea insists on promoting its alliance with the U.S. and its strategic partnership with China while respecting the one-China Principle."
Again, "strategic ambiguity." The One-China Principle, by the way, states that there is only one China, of which Taiwan and Mainland China are parts. At some point in the future Taiwan and China will be reunited into one country - One China - though how and when this will occur is anyone's guess. The government of Mainland China will constitute the government of this "future China."
South Korea has to worry about China more than Japan does. Let us not forget that they're also neighbors with North Korea, a country which they were supposed to reunite with at some point. South Korea is also one of Taiwan's chief economic competitors, and I can't blame them that much for trying to play both sides against the middle.
Sources were much sparser for the Taiwan-Philippines relationship. Most reflected the trade relationship to the exclusion of political relations which may or may not exist between the two countries. I did find a memorandum from the Philippines government, dated March 2018, which commands:
"Strict observance of the Philippines' One China policy commitment and unofficial people-to-people relations with Taiwan."
Translation: the Philippines would like continued access to Taiwan's markets, but would rather not anger China by recognizing Taiwan as an independent nation. Taiwan and the Philippines have never had good relations, so this comes as no surprise.
Lastly there's the U.S. point of view. This is from the U.S. Department of State's Fact Sheet:
"The U.S. and Taiwan enjoy a robust unofficial relationship...
...in the joint Communique, the United States recognized the People's Republic of China as the sole legal government of China, acknowledging the Chinese position that there is but one China and Taiwan is part of China...
"The United States does not support Taiwan independence... The 1979 Taiwan Relations Act provides the legal basis for the unofficial relationship between the United States and Taiwan, and enshrines the U.S. commitment to Taiwan in maintaining their defensive capability...
The United States insists on the peaceful resolution of Cross-Strait differences, opposes unilateral changes to the status quo by either side, and encourages both sides to continue their constructive dialogue on the basis of dignity and respect."
Sounds pretty straightforward, right? Except it's not. The U.S. has "influenced" Taiwan's national development at several key points, and claiming neutrality after the fact is disingenuous, if nothing else.
There's also that bit about "maintaining their defensive capability" in the above text. Who would Taiwan be defending itself against, I wonder? And why has the U.S. been sending military equipment and military advisors to Taiwan for decades? At the end of the day the U.S. wants Taiwan separate from China, and its actions demonstrate that. Keeping Taiwan separate allows the U.S. and other regional powers to contain China (a process often referred to as "encirclement"), frees shipping lanes, and blocks China's access to the Pacific. A China with open access to the Pacific means further instability in the region, and nobody - maybe not even China itself - wants that.
So is Taiwan a country? I would say yes, it is. Does everyone recognize that fact? Certainly not. But... if people fail to recognize (or choose to willfully ignore) certain facts about the world do they cease being facts? Of course not, and even less so as these facts become increasingly obvious.
*The Chinese Wikipedia entry for 國家 is linked in this paragraph. It does mention linguistic differences between "country/nation" and 國家.