WolframModelEvolutionObject properties are not listable. You need to map over each of them:

eo = WolframModel[{{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, 
      w}}, {{0, 0}, {0, 0}}, 3];
In[] := eo /@ {"VertexCountList", "EdgeCountList"}
Out[] = {{1, 2, 4, 7}, {2, 4, 8, 14}}

What is the version of Mathematica Your are operating with?

For 12.0 download WolframModel notebook and retry Your inputs after executing this source notebook in a copy of this notebook.

At first sight the given input is an example from this source notebook.

From the source notebook read the definitions and discover that

eo2[ {"EdgeCountList","VertexCountList"}]

does not work. It is not defined. A tranfer from the knowledge would have been

eo2[ "EdgeCountList][VertexCountList"]

but this results only in a correct output for the first property.