Yes, at least half of the exhaust is coming straight out of that gap without passing through the muffler. As you've noted, this leads to an increase in noise (or music, depending on your taste).
To my eye, that looks like a hassle to fix on your own. There appears to be the rusted remnants of two bolts holding the Y-pipe to the passenger side muffler and perhaps the merest wisp of a gasket remaining. I suspect that you're going to have to cut and drill out those old rusted bolts just to disengage the two pipes. A repair will either require those old bolts out, clear holes, clean mating surfaces, a new gasket, new bolts and lots of anti-seize. None of that is expensive but it's something of a drag.
If you've been thinking about an aftermarket exhaust system, this might be a good time to make a purchase. I'd suggest looking for a system that includes both pipes leading to the mufflers and the Y-pipe leading back from the catalytic converter.
This picture summarizes nicely why I like stainless steel in all exhaust components.