The AI nations do not ignore manpower constraints, they just deploy more divisions when you declare war on them, as you can as well. Economy does matter, it's required to resupply divisions. Divisions will wait to go into a battle unless they are fully supplied, otherwise nothing will happen (as per why the units have movement lines but don't damage each other). You can refer to the tutorial for more details.
I do agree with you however that this is not communicated well outside of the tutorial, so I will work on improving that.