This claim utilizes a true statement to feed readers a lie. It was confirmed by Press Secretary Karine Jean-Pierre here in a press briefing that president Biden will not be administered a cognitive test as a part of his yearly physical. The second half of the claim, that President Biden is the first president to not take such a test during a yearly physical exam, is false. As far as I can find that information does not exist, at least not in any official capacity released by a government entity.
I think this claim was made intentionally and with bad intentions. I think its not an accident that the true part of the claim is referring to The President's age and mental sharpness, from my understanding this a political issue which concerns voters across the political spectrum in the United States. Here is a ABC news / Washington Post poll from September which shows that. I think this claim was chosen because it has broad appeal, and gets reader's foot in the door so to speak. Many people already doubt Biden's ability due to his age, and the white house confirmed he will not be administered a cognitive test, so the lie about Biden being the only president to not take one is easy to accept without a second thought.
When you look into the two twitter accounts associated its seems clear that this claim is satirical at best, but likely intentionally misleading. The first account, WhaleWire, posts mostly about crypto finance, which is frankly a known grift at this point. There is also a website and mailing list by the same name which covers much of the same information and occasionally makes political claims like the one above. All of the written stories on this website are either not credited to an author, or credited to a "Mr.Whale" which is hyperlinked right back to the WhaleWire twitter account. To me, this suggests that the website is likely heavily AI generated, and it would not surprise me if the twitter posts are too.
The second account has a little more to it. It is also a Twitter account linked to a website and mailing list by the same name, this time Clown World Today. The website's about page describes itself as: "A newsletter for the 'conspiracy theorists' and others labeled 'dangerous' for speaking the truth and promoting freedom." So it seems pretty clear to me that this account is not concerned with truth so much as signal boosting claims which support it's already held worldview. The primary author for the website is a person identified as Ben Kaufmen, who's personal twitter account is linked on the Clown World Today website. His personal account is much of the same conspiratorial posts, but Kaufmen also posts a lot about crypto, which I think links him back to the WhaleWire account.
Ultimately, I think it is likely that both the Clown World Today and WhaleWire accounts and websites are run by the same person, maybe Ben Kaufmen. The content is similar in nature, and obfuscating the origin of claims like this is a tactic used by individuals and organizations to easier spread disinformation. So Biden is not being tested for mental acuity, but it is inaccurate to say he will be the first to do so. While the full intentions of the person behind the claim are unclear, the pattern of obfuscation and use of deliberate rhetorical tactics points to this claim being intentional disinformation from a bad faith actor.
