Typically, the "exhaust" on a WRX is two sections = a downpipe (bolts to the back of the turbo and runs down under the car to about the middle) and a cat back (a.k.a. midpipe + axle back = connects to the downpipe and runs all the way out the back of the car, including the muffler/tip). Together, the downpipe (dp) and cat back (cb) make up the turbo back exhaust (tbe). The downpipe is an aftermarket piece you should buy - bellmouth style downpipies are arguably the best (research). A catted vs. catless downpipe is something you need to decide for yourself - keep in mind that an aftermarket catted downpipe would only have one catalytic converter where the stock downpipe has two (research). If you're only worried about the cat back portion, sure, any exhaust shop could make one for you. It's a matter of deciding what size pipe (3" diameter is the most common), welding on a flange (possibly with an adapter if you're connecting a larger cat back to the smaller stock downpipe) to bolt it to the downpipe, bending everything to shape, welding on a few hangers, deciding what type of muffler/tip you want, and that's pretty much it. A downpipe is for performance and a tune is highly recommended/required if you replace it, while a cat back is mostly for looks/sound and can be installed any time without a tune and, like Chris said, just about any cat back (premade aftermarket or custom fabbed) will give you the signature Subaru rumble you're looking for.
Any time you take apart any section of the exhaust, it is highly recommended that you use brand new gaskets.
A less expensive, but still pretty good quality downpipe or cat back to consider is Invidia.
Note: Regarding the picture, the stock downpipe does have two catalytic converters in it. On an 02-05 WRX, they would be considered the "second" and "third" because there is a catalytic converter in the stock 02-05 up pipe (which is located just before the turbo). On an 06+ WRX, the stock up pipe does not have a cayalytic converter, so those would technically be the "first" and "second" catalytic converters.
The exhaust is just a pipe, but there really is a lot to consider if you're going to change any part of it. Good luck!
07 WRX (Albins 5MT + Blouch 20G)
Jorge Carrillo Tuned (365 whp/340 wtq)