A matrix is a subset of a tensor and the 4x4 matrix is absolutely a tensor. Tensors can be way more complex and do more but the 4x4 matrix you use in 3D operations is a great starting point.

A matrix is an array of numbers. It usually represents a linear map (in some basis). A tensor is more like that linear map.

In short, tensors generalize matrices. While we can probably guess accurately what "4x4 matrix" means, "4x4 tensor" is missing some information to really nail down what it means.

Interestingly, that extra information helps us to differentiate between the same matrix being used in different "roles". For instance, if you have a 4x4 matrix A, you might think of it like a linear transformation. Given x: V = R^4 and y = Ax, then y is another vector in V. Alternatively, you might think of it like a quadratic form. Given two vectors x, y: V, the value xAy is a real number.

In linear algebra, we like to represent both of those operations as a matrix. On the other hand, those are different tensors. The first would be a rank-(1,1) tensor, the second a rank-(2, 0) tensor.

Ultimately, we might write down both of those tensors with the same 4x4 array of 16 numbers that we use to represent 4x4 matrices, but in the sort of math where all these subtle differences start to really matter there are additional rules constraining how rank-(1, 1) tensors are distinct from rank-(2, 0) tensors.

"4x4 tensor" is not a useful notion. A (4, 4)-rank tensor would make more sense...