120v receptacle measuring 220v
As you've noticed, someone has wired this 20A 120V receptacle for 240V at 30A.  That is very dangerous, and it's quite lucky your house never burned down.  Most likely your old AC was designed to accept either 120 or 240V, and so it didn't notice the difference.  

Based on your pictures of the current wiring, I believe you have 10 gauge wires (wire looks like orange sheathed NM) which can handle the 30A it's breakered for.  It would be wise to double-check this with a wire gauge before upgrading to a larger air conditioner though.  

To replace the current unit with one that requires 30A of 240V, assuming you already have 10 gauge wire in the wall, you'll only need to change the receptacle to the correct one, such as a NEMA 6-30.  Or, if your new AC wants 20A of 240V, use a NEMA 6-20, and also change the breaker to a 20A one.