Sorry - MeshLab thought the model was manifold. Blender did find non-manifold areas however. I think Blender (and Shapeways) may use less precision than MeshLab, so vertices that are very close get counted as duplicates. (When checking this in Blender, first select all vertices and remove doubles, preferably with the difference limit set to zero rather than the default of 0.001, then
deselect all vertices before using "Select Non-Manifold", because it won't clear the current selection.)
I removed non-manifold vertices in Blender, then used the "Close Holes" feature in MeshLab to patch them, then the "Fill Holes" dialog in MeshLab to put "trivial" patches on several remaining small holes. Some of your whiskers got a bit shorter in the process, but otherwise I didn't see any problems, and Blender now thinks it is manifold.
I checked it in netfabb Studio Basic for flipped normals (which sometimes occur while patching holes), and it found none. It gives the volume as 42.9 cm3. I believe you have several overlapping meshes (or shells). Shapeways (and also netfabb, I think) computes the volumes of the shells separately, then sums them, without adjusting for the places where they overlap, so the computed volume may be quite a bit larger than the actual volume of the model (although your model is actually fairly large, by Shapeways standards).
I hope this helps.