I've never used Photo Mechanic so I cannot speak with athority on that but there are applications from two different companies. I am not surprised that they do not speak to each other. FWIW, ratings that one gives in Bridge do not show up in Lightroom either (and vise versa).
Let's say that before you import files into LR, you review them in PM and add stars and keywords (is that what you mean by "tags"). Then you import the files. LR should then read stars and keywords and other metadata. From my experience and a quick test, that's reliable.
Now imagine you import the files into LR, then add stars etc in PM. In this case LR doesn't know that you've done this and doesn't automatically update. You'd need to use Metadata > Read Metadata from files. But if you've done stuff in LR, this would overwrite it, and I wouldn't recommend using PM if you first import files to LR - you're just making work for yourself.
The above may explain what you are seeing. But are you sure that the files you are changing in PM are the same as the files in LR - you may have copies. Also consider that the Import option for embedded previews gives you much faster review speed and means less need for PM.