N dimensions can give you 2^N degrees of freedom.

If you have 3 spatial dimensions x, y, z, then you can have 3 degrees of freedom by translation and scaling along those dimensions. You can also have another 3 degrees of freedom by rotation around planes x^y, x^z, and y^z. I'm not sure what degrees of freedom are represented by the dimensionless scalar and pseudoscalar x^y^z, but probably one of them is mana points, and the other is hit points.

Two spatial dimensions give you two translation/scaling axes (x, y), one rotation (x^y), and one stamina points (scalar).

A 4x4 transformation matrix has 16 values, suggesting 4 dimensions, but it doesn't have 16 degrees of freedom, because several values are constrained. Nine values have three degrees of freedom, three other values have three more, and four are fixed, with zero degrees of freedom. Those six degrees of freedom suggest three dimensions, where two of the potential degrees of freedom are ignored.

So a higher-dimensional mathematical space can embed a lower-dimensional model, if the constraints are clever enough, and map the degrees of freedom onto the right dimensions or combination of dimensions. This might be useful for generating smoother animations using linear interpolation, or avoiding gimbal lock, or attempting to unify gravitation with electronuclear forces, or something else.

The difference is that people don't read the titles of press releases with pre-conceived notions of what a degree of freedom is, so it's not as useful for making clickbait.

5D has been a term of art for years, and was not made up by this article. https://en.wikipedia.org/wiki/5D_optical_data_storage

I don't know the optical storage space one way or another, so I was curious to learn more. As far as I can tell, that Wikipedia page you've linked to is solely about the same work by the same researchers as the article. Am I overlooking something? Has there been work by others using the same term? Or is the idea that the process of becoming a Wikipedia title is what makes it a term of art?

Even the Wikipedia page contains information about the original Tokio research team that only has one shared author. Furthermore, 5D is the natural extension of 3D, which was already used to distinguish it from traditional 2D (like, say, CDs).

"Term of art" might be a subjective concept, but the article making it up or not is objective.