Molecular similarity using Tanimoto coefficient
I was wondering if this means that the measure of distance is not for
  the molecule itself but for its representation?


Inherently, you must decide on a representation of a molecule in cheminformatics. In most cases, the representation is a "2D" valence-bond connection table (e.g., SMILES, SD file, ChemDraw, etc.) familiar to anyone who has taken an organic chemistry class.

So yes, a similarity measure depends on the representation. Tanimoto coefficients are usually based on comparing binary fingerprints.


  If so, is there a measure that compares two molecules based off of
  their inherent structure rather than the particular representation of
  choice?


I'm not sure how to answer this question since a representation forces choices about an "inherent structure" of a molecule. If I pick SMILES, I have no coordinates, implied hydrogen atoms, and choices about implied bond orders aromaticity, etc.

I think perhaps your question is about fingerprints - that "Is is there a similarity measure between a molecular graph - without generating fingerprints as an intermediate step?"

In that case, I think graph similarity measures are what you want, e.g. from Peter Willett, "RASCAL: Calculation of Graph Similarity using Maximum Common Edge Subgraphs" The Computer Journal (2002) pp 631–644

The reason people use fingerprint similarity measures is that they're easy to implement, and have worked fairly well for a wide variety of tasks.