I had the exact same error ("geometry is not m aware") with the program ("Polygon Neighbors") and weirdly this thread was the only one I found online about it. I was also able to fix it for me, although I don't know how general my solution is since it has nothing to do with the error message.
Anyway, here's what worked for me. Instead of listing the output directory as the one I wanted, I left it as the default ("C:\Users\corydeburd\Documents\ArcGIS\Default.gdb\WestBengal_ACon_PolygonNeigh2" for my folder system & specific file). I was then able to get the table. So, it had nothing to do with "m awareness" for me.
For some reason, there was an unspecified error when making this dbf so I just had to copy the whole thing (select all & Ctrl+Shift+C).
Godspeed all others coming to this thread with the same problem, I truly hope my solution works for you as well. I don't know why ArcMap is so ridiculously buggy, this sh** cray.