Element-wise multiplication of matrices with different dimension

You may Compile the code into a listable CompiledFunction as follows:

cf = Compile[{{a, _Integer, 2}, {b, _Integer, 2}},
   a b,
   RuntimeAttributes -> {Listable},
   Parallelization -> True
   ];

Here is a comparison

n = 100000;
m = 16;
mat1 = RandomInteger[10, {n, m, m}];
mat2 = RandomInteger[10, {m, m}];


Result = Table[mat2*mat1[[i]], {i, 1, Length[mat1]}]; // RepeatedTiming // First
Result2 = cf[mat1, mat2]; // RepeatedTiming // First
Result == Result2

0.342

0.067

True