I should start off by saying that I am biased and haven't owned a Mustang. I love the AE86 and have owned several of them. In my somewhat limited experience with other cars often compared to the AE86 (Mustang (it was actually a Fox-body Capri,) first-gen RX-7, S12 Nissan 200SX) the main thing the AE86 has going for it is that it is pretty well developed for the type of driving we like when compared to some of the others. Toyota basically mass-produced a car like the modified Mk2 Escorts being roadraced and rallied around Europe. It has an optional multivalve twincam, steering rack positioned well to minimize compromises, similar suspension layout, similar size, weight, track, and wheelbase to the Escort.
The Corolla isn't super-stiff, especially by modern standards, but it took me a combo of sticky tires, aggressive driving, and bumpy surfaces before I started thinking about chassis reinforcement. AFAIK, chassis flex is a concern with Mustangs, especially when you take extra torque, and bigger tires into account. I'd give the Corolla a nod here unless the Ford faithful want to correct me.
The Corolla has a solid rear axle located by four parallel trailing arms (two upper, two lower) and a panhard rod. The Mustang has a solid axle located by two parallel lower trailing arms and two angled upper arms. The Mustang design has a higher roll center, and a more steeply inclined roll axis than the Corolla, and roll bind becomes a problem with the Mustang design sooner than the Corolla design, particularly when both cars have stiffer-than-stock suspension bushings, and the result is snap oversteer. Short of bind, the Mustang rear design also tends to overload the outer tire under cornering sooner, increasing understeer.
Both cars need roll-center correction at the front if lowered, because lowering either one will drop the front roll center too low, and makes bump-steer issues worse. AFAIK, Corollas don't have the same Ackerman problems that Mustangs do out-of-the-box.
To make a long story short, neither car is has perfect handling out of the box, but the Corolla offers a slightly better executed solid rear axle suspension setup and lighter weight than a Mustang. The Corolla has good handling and is fun to drive in stock form. I'm sure the Mustang could offer a similar driving experience with a bit of work on the suspension.
The Mustang aftermarket is huuuuuge and offers all kinds of help with the rear suspension, with some solutions as simple as careful bushing selection on the stock rear arms, to some that are as involved as ditching the angled upper arms and going to a three-link plus Panhard or three-link plus Watts linkage setup. There are front suspension upgrades as simple as springs and shocks, or as involved as a new front subframe with improved geometry, better adjustability, and repositioned steering.
Get a Corolla if you can; they are fun, fun cars and can grip well or go sideways well. They respond well to skilled driver input, whatever your preferred driving style. They are good in stock form and great when lightly and sensibly modified. If you end up getting a Mustang, I am sure it will be a fine project car, and will also be a lot of fun. Don't let the import community get you down, and don't get so hung up on suspension theory that you don't ever get out there and enjoy it! It will probably show a bigger percentage improvement than the Corolla with light tuning, and should have better parts availability, bigger aftermarket support, and you don't have to worry about how to fit tires under it as much. (My 195/50-15s rub on my lowered Corolla!)