L293DD overtemperature protection
The answer is, if you want a good design, then NO. You can't rely on over temp protection. Whenever you go beyond the specs in the datasheet especially the maximum ratings all bets are off. The engineers designed this part with 1.2Amps in mind and also tested it with 1.2Amps. You don't know why they did this. Maybe they had 1 out of 100 blow up when they tested it beyond 1.2 Amps, maybe they hit 90C and 1.2Amps and then they start to die. maybe its just an intermittent problem. You could be running your chip and it could fail at any time. This question is like saying can I run my car engine at 7000RPM if the red line is at 6000RPM? Yes you can, do you want to no probably not.

There are a few ways you may be able to overcome this problem.
If your worried about an overtemp condition, put a heatsink on it. Even a piece of aluminum that is 2 or 4 times more area than the IC can help drive the temperature down by 10's of degrees.

You can parallel the devices and half the current going through the IC. Usually this works with TTL and BJT which is what the drivers say they are.