Dryer Outlet - Two 120V's testing fine. No 240V
Answer pending verification of photo...

It looks like someone may have been using that dryer outlet for a large 120v appliance, probably with some homemade adapter.  In your picture it seems like both the red and black wires are attached to the upper screw on that double pole breaker.  This would put them both at 120v to ground, but zero volts between each other.

Having parallel paths in a circuit like that is not allowed, but it doesn't matter much now because you're going to fix it.  Once the two wires are attached to the right screws you should be fine.  Just to double check that there is not an issue with your panel, measure the voltage between the two screws on the breaker.  That should be 240v, of course.  If it's not, there could be a compatibility issue with that breaker in that panel, but like you say, the stove works, so I don't think it there would be an issue here.  Turn the breaker on and off to reset it if the voltage is not present.