Looks like it's all been said... but I'll throw in my $0.02 worth.
To the original question, yes, undoubtedly, you can make a stand out of plywood that doesn't require a dimensional lumber frame. I've done so, many times. With good materials, glue, and joinery techniques, you can far exceed the strength of a 2x4 pine stand.
OSB or MDF are no-no's in my book. Why? If the sealer coat fails, they're mush. I have an MDF top on my reloading bench, sealed with poly. Been using it for 15 years or so, no problems... BUT. It's not carrying a huge glass box full of water, either. The amount of damage a collapsed aquarium stand is capable of doing to your home has to be factored in... something that will retain some strength if it happens to get wet would seem the least precaution you could take. I know, others do it... that's their choice, but I won't be using it in my home. Edge finish on ply? There are many techniques that work fine, from as simple as iron on edge banding, to framing with solid wood... to cutting joints at a 45 so there is no edge grain showing. It's not that difficult to come up with something acceptable.
Pocket screws seem to be all the rage among the plywood builders these days. It's a fine technique, but it's not the ONLY technique you should consider. Look at each joint, what kind of pressure it's going to be under, and decide on a case by case basis. Biscuit joints are what I typically use, rather than pocket screws. Stronger, and simple, but you do have to have the right tools. Dado and Rabbit joints also seem to find their way into any plywood stand I've ever made, as do butt joints backed with hardwood glue blocks.
With all that said... my latest stand is welded, powder coated steel. Why? Cheap, strong, and allows for a very open design. I'm probably done building large wooden aquarium stands. In my opinion, steel is simply a better material for this purpose. The downside, of course, is that you have to either have the skills and tools to work with this material, or be willing to pay someone else to do so. I had a small fab shop build my steel stand to my specifications, cost me less than what the lumber would have cost to do it myself.