The 1 by 2s on the subfloor is mainly to give you something to screw into when you fasten your sidewalls to the subfloor.
(Yes, it also gives you a place to insulate the bottom if that is what you want!)
The 1x2's are securely glued to the plywood floor, and the floor subassembly should be bolted to the frame, through the plywood and through the subfloor.
It shouldn't go anywhere.
Of course if you want to build a tank, you could always go to a 2x2 subfloor, or even a 1x3 like you mentioned... (I believe I used 1x3s on my widget subfloor...
Mike...