Meanwhile it seems to me that the discrete analogues to all Stiefel-Whitney numbers of $d$-dimensional manifolds are invariants of this type:
First, for every $n$ there is a rule to color every $d-n$-simplex of a simplicial complex (with branching structure) describing a $d$-manifold by a $Z_2$ element such that
- the color or each simplex only depends on how the simplicial complex looks in a small patch around it.
- the coloring forms a $Z_2$ $d-n$-cycle in the simplicial complex.
- this cycle is a combinatorial representant of the $n$th Stiefel-Whitney class of the $d$-manifold.
Second, there is another rule to $Z_2$ color every $d-n-m$-simplex of a simpicial complex given a $n$-cycle and a $m$-cycle in the simplicial complex such that
- the color or each simplex only depends on how the simplicial complex and the two cycles look in a small patch around it.
- the rule is such that the coloring is a $Z_2$ $d-m-n$-cycle.
- this cycle is a combinatorial representant of the cup product of the two cycles.
Such rules are for example given, at least partly, in https://arxiv.org/abs/1505.05856.
Now a Stiefel-Whitney number is an integral over a cup product of different Stiefel-Whitney classes. So combinatorially we can combine the two rules above to get a $0$-cycle and then count $Z_2$-colorings of the vertices mod $2$. So this corresponds to an invariant of the above form if we choose as "kind of object":
"Every vertex that is colored by $1\in Z_2$ via the combination of the two rules above."