They're not being precise with their language, but their point is largely true. What they really mean is that the US has the oldest still active Constitution in the world. The UK has existed in a continuous government for far longer, but they don't have a written Constitution like the US does.
Yeah, it's easy to shit on Americans about being ignorant of history. But this person's point is largely true. The US has had the same constitution in effect for nearly 250 years. It is the oldest written constitution on Earth still in effect. Most nations have revolutions or complete rewrites of their foundational legal documents long before they reach this point.
And this is also why the US has such political instability right now. We have a Constitution that was written for the needs of 250 years ago. It was formed from a series of compromises that made sense in the politics of 250 years ago. At this point, we really should scrap it entirely and start from scratch. Having the world's oldest Constitution really isn't something worth bragging over; it just means you're running obsolete software.