>A theory of the type you are asking cannot be **that** concrete, because:

1. By an old result of Victor Marek, it is consistent with the axioms of $ZFC$ that the second order theory of every *countable structure* (in a countable vocabulary) is categorical. See this [FOM-post][1] of mine for a reference.

2. In the above FOM-post, I conjectured that Marek's result can be extended to all *Borel structures*. This conjecture was verified by Harvey Friedman in this [FOM-post][2].

3. In yet another [FOM-post][3], Solovay showed that it is consistent with $ZFC$ that as soon as a second order theory $T$ is both axiomatizable and complete, then $T$ is categorical. See also this other [related FOM-post][4] of Solovay.


  [1]: http://cs.nyu.edu/pipermail/fom/2006-May/010544.html
  [2]: http://cs.nyu.edu/pipermail/fom/2006-May/010545.html
  [3]: http://cs.nyu.edu/pipermail/fom/2006-May/010549.html
  [4]: http://www.cs.nyu.edu/pipermail/fom/2006-May/010561.html