What is the proper usage of "identically zero"?

You are correct - when we say something is "identically" some value, generally we are talking about a function taking on that value everywhere. So it doesn't make much sense to say that a matrix is "identically zero". Though it is possible to think of a matrix as a linear transformation, in which case you can say that the linear transformation associated with the matrix is identically zero. This is true if and only if $A$ is the zero matrix.

The phrase "identically zero" is generally used when we need to distinguish between a function having a zero at some point and a function being the zero function. Either might be written $f(x)=0$, for instance, so it helps to have a way to distinguish the two cases.

In the case of a matrix I'd just say the matrix is nonzero or whatever.

Usually this would entail that there's some sort of dependence going on, that some of the entries of $A$ are functions, and while for certain inputs, it's alright that $A$ zeroes out, you are trying to avoid just having $A_{ij}(x) := 0$