I don't think I've seen that term used before, but a brief search on the web on those terms confirms my initial thought.
"Matrix notation" is when you have selected a particular basis (bases) for the vector space (spaces) upon which you're working, and matrices are written as a 2 dimensional array of coefficients with respect to those bases.
"Abstract matrix notation" is when you have not done the above.
For example, suppose you premultiply a vector by a matrix to yield another vector.
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