No, not really. Symmetries of an n-gon (a 2-dimensional figure with n sides) are just permutations of the n numbers representing the sides. A cognitive metaphor is to call certain types of symmetries “rotations” or “reflections”, because we can’t really describe all symmetries through a metaphor like that. The symmetries are not literally operated by picking up your figure and flipping it over.

On the other hand, mathematical abstractions of dimensionality not cognitive metaphors. A “hole” is a meaningful topological abstraction in various dimensions that corresponds to what we consider a colloquial “hole” in 3-dimensional space. If I tell you a Klein bottle is a bottle that has only one surface (no inside or outside) without any “hole”, I’m not making a metaphor, I’m describing a 4-dimensional object as closely as I can describe a mug or a vase in 3-dimensional space. Just because I’m not defining it through pure mathematics and we can’t immediately visualize it (in entirety) doesn’t mean it is a cognitive metaphor, because it perfectly corresponds to the actual concept of dimensionality (instead of being a clever figurative description of it).

Similarly, if we geometrically represent a black hole we can clearly see the curvature of space (from one angle), leading down to an event horizon, within which is a singularity. This representation corresponds to rolling a coin at an angle down a curvature, watching it spin at the bottom a bunch of times, then fall in. In fact, a shuttle falling out of orbit down to the event horizon would look just like this as it entered the whirl zoom of a black hole, before the unstable orbit failed and it just fell in.

Even further, there is a very cogent, topological sense of what “within” means here. Just as you can fall into a manhole in the middle of the street, you can fall into the event horizon of a black hole. If we represent a street as a 2-dimensional plane, a manhole in the street is the topological space in the plane that has no points - a mathematical hole. In three dimensions, a jelly donut has a topological hole, and the hole is also 3-dimensional (the missing points would be representable as 3-dimensional vectors). Information theoretically speaking, the inside of a black hole’s event horizon does not exist - it’s strictly an absence that we can only reason about heuristically. It would be a cognitive metaphor for me to say that we can only reason about the inside of the event horizon in the same way we can reason about the middle of a donut that tastes like nothing because there is “no donut”. But it is not a cognitive metaphor to call the black hole a “hole”, because fundamentally and literally it acts like one. The singularity inside the black hole is not a hole, but that’s different.

Our common sense of what it means to be a hole is literally described by the mathematical abstraction of a hole. When mathematical abstractions are actually just literal abstractions of what we already understand, they’re not cognitive metaphors. If you’d like an example of a cognitive metaphor insead of a pure abstraction, look at string theory. While our common sense of a hole literally corresponds to its 3-dimensional topological abstraction, our common sense of what we call a “string” has no precise mathematical correspondence. A string in the common sense of the word is a 3-dimensional object that is very long and thin. A string in the mathematical/physical sense is a one dimensional object, like a line. To call it a string is to invoke an intuition of something like a small thread that in some sense seems “barely” 3-dimensional.