Sure, "5D" _could_ make sense. It just happens not to in this case.

They're mixing up two different sorts of dimensionality that it doesn't make any sense to mix up.

1. How many parameters does it take to describe where you're looking for a given piece of information?

This is (kinda) a measure of how many pieces of information there are. Increasing the number of dimensions can mean a very large increase in the amount of data -- imagine going from a 1000x1000 array of things to a 1000x1000x1000 array.

The number of dimensions in this case is "two and a bit", because they're using a small number of 2-dimensional layers.

2. How many parameters does it take to describe what you find when you look at a particular piece of information?

This is (kinda) a measure of how much information there is in each thing. Increasing the number of dimensions here typically means a rather small increase in the amount of data. E.g., the difference between "1D" and "2D" here would, in the simplest case, be a factor of 2.

The number of dimensions in this case is 2, because in each voxel there's a thing parameterized by size and orientation. (I think.)

So, three dimensions of the first kind and two of the second. Add 'em up to get 5D, right?

Wrong. (At least, if your aim is to inform rather than to deceive.)

First problem: these two kinds of dimension do not behave in the same way. Suppose there are N positions along each dimension, and you have A type-1 dimensions and B type-2 dimensions. Then you have N^A places to look for data, and each one stores B log N bits. So adding 1 to A multiplies the amount of data stored by N, typically a very big change; but adding 1 to B multiplies it by (B+1)/B, typically a rather small change.

Second problem: the "baseline" is not zero type-2 dimensions but one. You always have at least one type-2 dimension, or else you aren't actually storing any data. So e.g. if you have a grid of memory cells, that's 2 type-1 dimensions and 1 type-2 dimension. Are we going to call this 3D storage? I really hope not.

Two-and-a-bit-th problem: as mentioned above, their third type-1 dimension is really hardly there: they have three layers. I guess that is a third dimension, kinda, but it's pretty underwhelming.

This is 2-and-a-bit-dimensional storage, with each storage location being able to hold a bit more data because of this size-and-orientation thing. It sounds like it's an impressive technical achievement and might turn out to be a great storage medium. It just isn't in any useful sense five-dimensional.

I said at the start that "5D" could make sense. How? Well, suppose you have a system where you have things arranged in a 3D lattice, and then to query each thing you shine light at it and see what it does. And suppose you're able to make each of your thing store different things for different wavelengths of light, and also different things depending on what angle you shine the light at. In that case, the number of type-1 dimensions might be as large as 6: storage locations are indexed by three spatial dimensions, two dimensions describing the direction of the light beam, and one dimension describing the light wavelength. (If you had a system like this, then I bet that in practice those last three dimensions would all be "small", like the third spatial dimension in this silica-glass storage.)