Looks like a muffler bearing
If it is a muffler bearing, it is misplaced. It is too far forward.
Are the wires behind it going into whatever that is 
Zig
The wires behind it are not connected to it. They are for something else.
I think it has something to do with emissions control.
What year? Where are you?
Or: it could be an emergency hockey puck 
No wires or any other connection to anything but the bracket, so it can't be emissions related. The car is a '98 Sport, originally sold in Wyoming and registered in S Dakota, now in Arizona.
It could be a hidden emergency hockey puck--the vehicle was originally registered in one of our northern states.
Somebody got REALY confused while doing a body lift 
Zig
Maybe the extra weight was intended to lower it instead?
Maybe someone has plugged off a hole designed for a second O2 sensor. Later models have one behind the cat that measures its effectiveness.
There is no hole in the exhaust pipe. This thing is attached to a bracket welded onto the pipe.
There is a second O2 sensor ahead of this thing.
Bobzooki: I haven't found my dictionary yet, so I might have to agree with you.
I think the vibration damper suggestion probably makes sense. This thing is a round piece of steel attached with rubber to its mounting bracket, so it can move. Seems to me there would be a simpler way.
Maybe the exhaust pipe is short enough on the 2-door models that these are not needed.