Given the computational expense note, and assuming there is no form of BinLists that does this, I think the following does what you want, labeling the elements by bin.
altBin3D[list_] := SortBy[Floor[#[[1]]] -> #&/@ GatherBy[list, Floor], First];
In[6]:= altBin3D[Tuples[{0.5, 1.5}, 3]] // AbsoluteTiming
Out[6]= {0.000074, {{0, 0, 0} -> {{0.5, 0.5, 0.5}}, {0, 0, 1} -> {{0.5, 0.5, 1.5}}, {0, 1, 0} -> {{0.5, 1.5, 0.5}}, {0, 1, 1} -> {{0.5, 1.5, 1.5}}, {1, 0, 0} -> {{1.5, 0.5, 0.5}}, {1, 0, 1} -> {{1.5, 0.5, 1.5}}, {1, 1, 0} -> {{1.5, 1.5, 0.5}}, {1, 1, 1} -> {{1.5, 1.5, 1.5}}}}
A solution which fills in blank bins and doesn't label them, by constructing the list of bins and then using the labeled bins to replace them, is
bin3D[list_] :=
Replace[Flatten[
Table[{i, j, k},
Evaluate[
Sequence @@
Transpose[
Prepend[Through[{Floor[Min /@#] &, Floor[Max /@#] &}[Transpose[list]]], {i, j, k}]]]], 2],
Append[Floor[#[[1]]] -> #&/@ GatherBy[list, Floor], _ -> {}], {1}]
In[9]:= bin3D[Tuples[{0.5, 1.5}, 3]] // AbsoluteTiming
Out[9]= {0.000189, {{{0.5, 0.5, 0.5}}, {{0.5, 0.5, 1.5}}, {{0.5, 1.5, 0.5}}, {{0.5, 1.5, 1.5}}, {{1.5, 0.5, 0.5}}, {{1.5, 0.5, 1.5}}, {{1.5, 1.5, 0.5}}, {{1.5, 1.5, 1.5}}}}
but it looks to be as much as 1000 times slower (~.3 seconds vs ~.005 seconds for RandomReal[{-10, 10}, {1000, 3}]).
