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](http://mathworld.wolfram.com/GraphGenus.html). 

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

> [National Science Foundation Mathematical Sciences Institutes:  *Topology of Shapes, Persistent Homology and Point Clouds: Where Does it Take Us?*](https://mathinstitutes.org/highlights/topology-of-shapes-persistent-homology-and-point-clouds-where-does-it-take-us/)

> [H. Edelsbrunner and J. Harer. Persistent homology --- a survey. Surveys on Discrete and Computational Geometry. Twenty Years Later, 257-282, eds. J. E. Goodman, J. Pach and R. Pollack, Contemporary Mathematics 453: *Persistent Homology — a Survey*](http://pub.ist.ac.at/~edels/Papers/2008-B-02-PersistentHomology.pdf)