Can chemicals be identified by their molar mass?
It depends. 

This was pretty much the way that the elements were discovered. 

For inorganic chemicals in general the problem is really one of specificity. So I want to work with a chemical sample between 1 gram and 20 grams and I have an analysis good to 1%. That only yields about 300 statistically significantly different numbers $1.01^{300} = 20$. So for various inorganic salts this wouldn't work too well. 

To take this into another field think of geology. Hardness and specific gravity are simple tests used to help identify minerals. Neither is very good by itself, but combined they sort out the minerals into a lot more bins.