Earlier quoted context omitted.
I was confused as hell for a long time when I first got into ML, until I figured out how to think about tensors in a visual way. You're right: fundamentally ML is about vector and matrix operations (1D and 2D). So then why are most ML programs 3D, 4D, and in a transformer sometimes up to 6D (?!) One reasonable guess is that the third dimension is time. Actually not. It turns out that time is pretty rare in ML, and it…
For whatever reason, I have held a mental image of a Tensor as a Tesseract/HyperCube where the connections are like the Elastic workout bands where they have differing tensile resistances, and they pull on one another to create their encapsulated info-cluster - but I have no clue if thats truly an accurate depiction, but it works in my head.... https://upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Or...
The most straightforward mental model I've ever found for ML is, think of it as 2D matrix operations, like high school linear algebra. Matrix-matrix, matrix-vector, vector-matrix, and vector-vector will get you through 95% of what comes up in practice. In fact I'm having trouble thinking of something that doesn't work that way, because even if you have an RGB image that you multiply against a 2D matrix (i.e. HxWxC multiplied by a mask) the matrix is still only going to apply to 2 of the channels (height and width), since that's the only thing that makes sense. That's why there's all kinds of flattening and rearranging everywhere in practice -- everyone is trying to get a format like N,C,H,W down to a 2D matrix representation.
People like to talk up the higher level maths in ML, but highschool linear algebra (or for the gamedevs in the audience, the stuff you'd normally do in a rendering engine) really will carry you most of the way through your ML journey without loss of generality. The higher level maths usually happens when you start understanding how differentiation works, which you don't even need to understand until way later after you're doing useful things already.