First

In[2]:= $Version
Out[2]= "10.0 for Microsoft Windows (64-bit) (June 29, 2014)"

In[6]:= Assuming[Element[M, Matrices[{2, 2}]] && Element[Z, Matrices[{2, 2}]], Simplify[M == M Z]]
Out[6]= False

With this elementwise multiplication has been ordered. The result is almost always correct, with the exception of Z = {{1,1},{1,1}} or M = {{0,0},{0,0}} - over the complex numbers (which is the default field in Mathematica), so, with other words, the result is itself false. If matrix multiplication is ordered, the outcome is

In[9]:= Assuming[Element[M, Matrices[{2, 2}]] && Element[Z, Matrices[{2, 2}]], Simplify[M == M . Z]]
Out[9]= M == M.Z

In[10]:= Assuming[Element[M, Matrices[{2, 2}]], Simplify[M == M . IdentityMatrix[2]]]
Out[10]= M == M.{{1, 0}, {0, 1}}

Here Mathematica in fact states I don't know, which is the usual outcome for unspecified symbols:

In[11]:= a == b + c
Out[11]= a == b + c

In[12]:= a == b + c /. {a -> 7, b -> -1, c -> 8}
Out[12]= True

Equal[] returns True if lhs and rhs are identical. For M = M Z or M = M . Z this is unkown, because they do not evaluate to something.

You mean a check if assuming Element[M, Matrices[{2, 2}]] && Element[Z, Matrices[{2, 2}]] is then Element[M Z, Matrices[{2,2}]] or Element[M . Z, Matrices[{2,2}]] true? You cannot test this with Equal[]. A way is In[16]

$Assumptions = {M \[Element] Matrices[{2, 2}],  Z \[Element] Matrices[{2, 2}]};

Out[14]= {M \[Element] Matrices[{2, 2}, Complexes, {}], 
 Z \[Element] Matrices[{2, 2}, Complexes, {}]}

In[33]:= M Z // TensorDimensions
During evaluation of In[33]:= TensorDimensions::ttimes: Product of nonscalar expressions encountered in M Z. >>
Out[33]= TensorDimensions[M Z]

In[36]:= {{x1, x2}, {x3, x4}} {{y1, y2}, {y3, y4}} // TensorDimensions
Out[36]= {2, 2}

In[16]:= M. Z // TensorDimensions
Out[16]= {2, 2}

In[34]:= $Assumptions = {A \[Element] Arrays[{2, d, 4}],  B \[Element] Arrays[{d, d}]};
In[35]:= TensorDimensions[A\[TensorProduct]B]
Out[35]= {2, d, 4, d, d}

elementwise multiplication is not recognized assumptions-symbolically under Mathematica 10 ...

Yes, I think we agree.

Yes

I hope it was obvious that I was saying it should return "M == M Z", rather than "False"

Yes

There are solutions where M is not all zeros and Z is not all ones.

True - and that means, that the mentioned result is not almost always correct (as I said), because there is not a finite set of exceptions, but an infinite set of exceptions.