Basically what it says in the title. The title was actually intended to be:
"Can any 2D tessellation be uniquely determined by a partition of its vertices into types and for each type a sequence where the nth value is the number of cycles of length n containing any given vertex of that type?"
but that was too long. When I say cycle here I mean it in the graph theory sense - pretend the tessellation is just a graph. I have a hunch that this is the case, but I don't know.
Example: a square tiling of the plane has one type of vertex, characterized by being in no odd-length cycles, exactly four cycles of length 2 (edges), four cycles of length 4 (the square faces adjacent to the vertex), etc.
I feel like this, for each n, may be enough information to uniquely identify it across all possible tessellations of the plane or perhaps of any other surface - but I don't know that, and I wouldn't be able to guess how to prove it.