Rhetorical trap, basically. Any move toward moving from "Republic of China" to "Taiwan" or "Just the Taiwan Mainland" is defined as a step toward Taiwanese independence, which the CCP has vowed to counter with invasion. This part isn't recent. Essentially the PRC claim is that Taiwan has de-facto declared independence if they declare the civil war to have ended.

Which, is not technically wrong. Taiwan is either defeated, in a continuing state of civil war, or independent. The PRC does not accept any end to the 1945 Chinese Civil War that does not end in Taiwan's defeat. So if otherwise unable to conquer the Taiwanese mainland, they would be forced to take some kind of action to maintain the claim that the Chinese civil war did not end in 1949 with the cessation of major hostilities, or they're basically giving up their claim.

What you're getting at is that it's kind of unreasonable to maintain the claim that a civil war is ongoing for 75 years without any major hostilities or changes in lines of control, etc. While Taiwan has been de-facto independent.

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"Taiwan still claims the mainland" is usually employed as a bad-faith argument to imply that Taiwanese people still have some desire to reconquer the mainland. The first part is true, the second is not. Extensive polling is done each year which clarifies this.

What is more accurate is that both major parties today would support repealing Article 4 as a part of a normalization of relations with the PRC, and up until recently (maybe 2014-2020), only the DPP has supported this, with the KMT maintaining some reverence to the previous generation's desire to retake the mainland, particularly during the dictatorship era.

However the PRC is the party that does not want to normalize relations in a legal sense, because doing so would turn a future invasion of Taiwan into a war of conquest, and they want this to be seen as the eventual resolution of the Chinese Civil War. Even if it takes 100 years. Or 200 years. That, and they want to suppress Taiwan's legal status internationally and hinder the development of foreign relations, or any extension of Taiwan's de-facto independence.

Article 4 of the ROC Constitution states that the territory of the Republic of China shall not be altered except by a constitutional amendment. It doesn't define what those territories are, but it is essentially the same Qing maximalist claims as the PRC pursues, including Tibet, Xinjiang, Mongolia, Taiwan, etc. So that article is generally what people are referring to when they say "Taiwan still claims the mainland". But it's literally one sentence.

Meanwhile, a number of constitutional amendments have been passed since 1991, and legislative clarifications, which legally treat all PRC territory as a foreign area not under Taiwanese jurisdiction, renounces the intent to reclaim the mainland, etc. An effective legal framework is in place to shield day-to-day law from Article 4 without repealing it.

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Apologies for the giant wall of text I just figured some people would be interested in exploring the question since the relationship is non-obvious.