Yes. The difference is that here the operation represented by putting wires next to each other is the direct sum, whereas in the Penrose notation the operation is the tensor product. They're both examples of the string diagram notation for monoidal categories. (https://forum.azimuthproject.org/discussion/2335/lecture-73-...)

I expected it to -- graphically, it looks very similar to Penrose notation! -- but at first glance it seems to be quite different.

Penrose notation is very high level: edges correspond to tensor dimensions, joining corresponds roughly to tensor contraction. When you're working with multiple complicated multi-dimensional tensors, it can be a beautiful way to express things.

This notation seems to be expressing something a bit lower level: it seems that edges correspond to values and dots correspond to copy/add. So, roughly, they visually express linear functions. (I didn't read the full post, it's possible that they expand beyond this initial meaning.) Diagrams of a similar style are used in other contexts (eg. expressing the weights of neural networks), often as a pedagogical tool.

It might be easier to think of it as a kind of Feynman diagram: a sum over paths. That's what matrix multiplication is, it's a kind of sum over weighted paths. A matrix records all the ways to get from the set of "columns" to the set of "rows".

Yes