Academy of Sciences of Heroesland

But a sphere in 3D space is two-dimensional. It doesn't become three-dimensional.

Wikipedia in that article about a sphere disagrees with you.
Although, I read the article about the Poincaré conjecture again, and it's written the same way as you said.
But that's not the main point; it's about which terminology to use. I prefer to call the sphere three-dimensional, and the manifold on it two-dimensional, because, as I said again, to describe the sphere we are familiar with, we need three dimensions – it is a three-dimensional figure. Then the Poincaré conjecture would be formulated as follows: a simply connected compact n-dimensional manifold is homeomorphic to an (n+1)-dimensional sphere. But, of course, it's all just a matter of terminology.