Yep for switching operation that graph is just to be used to model the transition between the off and on state. By having no DC area on that graph means that you are not supposed to use it in its linear region because this causes only small parts of the transistors structure to turn on, get hot, go into a thermal runaway, get way hotter and blow up the transistor.
These transistors will handle 40A just fine, tho the the pair together will dissipate about 25W of heat, so it does need a fairly chunky heatsink to passively cool, but it is possible. Tho the PCB traces most definitely will not handle 40A. Perhaps up to 15A if you tin them. But you could possibly solder a big thick 4mm2 cable on top of those traces to beef them up.
Where the design might be lacking however are safety features. There is no current sensing of any sort. So if this is powered by a high current capable source such as a large battery then a short circuit will most likely kill two of the transistors in less than a second, if you then attempt to put the motor in reverse this kills the other two causing the whole bridge to die into a short circuit with full battery current flowing trough it until something melts and breaks it. This could be mitigated by sensing current in the ground return, feeding that into a comparator and disabling the bridge in the event of exceeding the current. Similarly trying to hold the MOSFETs permanently on with no PWM signal might cause the bootstrap capacitors to drain down to a point where the FET gets into a linear region, gets into thermal runaway and blows up. This could also be mitigated by a circuit that disables the bridge if the control signal stays high for too long.