Two answers 'by vague association'.
The graph invariant called 'genus'. A traditional, important graph invariant spiritually similar to what you are asking about is the genus of a graph.
I am aware that one may argue that this does not capture the intuition of 'dimension'. But then again, you yourself tagged this a 'soft-question', so this answer should be sort-of-acceptable to you.
Also worth pointing out: as you will probably known, for any fixed dimension $d\geq 3$, any countable graph can be embedded into $\mathbb{R}^d$, a fact which by the way can be proved working only over the 'signature' of graph-theory, plus a little intuitive geometry, by an inductive argument; you do not need the usual argument via a Vandermonde matrix (not that this would be 'bad', yet the proof via the 'moment curve' wields a 'signature' which besides $\sim$ (adjacency) and $\#$ ('intersects') also uses $\mathbb{Z}$, and '$+$' and '$\cdot$').
The methods around the ideas of 'thickening point clouds' and 'persistent homology'. Your question also reminded me of the very active field of 'persistent homology' (and related ideas). Again, it does not fit your specifications precisely. For this to be relevant to your question, of course, the data that you wish to associated your isomorphism invariant to must be given inside a metric space. A purely combinatorial graph will not be enough to make these methods 'bite'.
If so inclined, you can start reading about this in the following two references