Not for this kind of situation, no. Mostly because there's nothing technically invalid about coincident geometry. Invalid geometry is called "non-manifold," and there are tools for identifying it like "Select -> non-manifold." But even non-manifold geometry has its uses.
But the situation that you're looking at isn't necessarily invalid. In fact, for a game asset this might be a good way to limit polygon count (if you can resolve the coincident faces well).
The suggestions to study modeling practices are good suggestions because a non-trivial amount of good modeling practice is avoiding problems, as much as it is understanding how to solve them.
To avoid being unhelpful, though, there's not really a great answer to your particular situation. There's no tools that will detect where cuts need to be made so that you can connect those two meshes. No matter what solution you choose, it will necessarily involve some amount of guessing about where to cut the horizontal piece, and then manually attaching the vertical piece. Even separating them into different objects so that you can Bool them together is problematic because you'll still have to manually tweak the Bool result.
However it might not be a problem worth solving. The only problem with coincident faces is when they are exactly coincident. If they are mathematically in the same place, the renderer has to "guess" at which one is in front of the other. Computers are (at best) terrible guessers. They are, however, excellent precision...ers at precision. Which means the faces only have to be slightly separated in order for the renderer to know which one is in front of the other.
So, practically speaking, I'd recommend scaling your vertical pieces up (or down) by 0.0001. (You can do this in the numeric panel by manually entering the scale value in all three fields, or if you're comfortable with "ghost input" you can press s and then just type "0.0001" and hit enter). That'll be just enough for them to overlap the horizontal piece so that the renderer can differentiate them. But it'll be a small enough amount that the human eye won't really notice... it'll still look glitchy in the viewport, though.
Hope that helps!