I'm not that familiar with DoaM as my group decided they didn't like 4e so we never had to deal with it. Is it something that all classes and monsters have or is it something limited to only certain characters? Does it apply to all attacks all the time or just certain types of attacks?
To me it kind of sounds like a lot of save for half spells. Is that an accurate comparison? Is the damage on a miss a set amount or does it depend on what the damage for a hit would have been?
It seems that the main argument against it is it is not realistic. Is there a mechanical reason people don't like it for instance is something people saw abused in some way?
These have mostly been covered by [MENTION=1927]Thaumaturge[/MENTION].
In 4e, most daily powers are either Reliable (= not expended on a miss) or have some sort of effect that occurs even on a miss. For many power in the latter category, that effect is half damage. Which, as you say, is 4e's functional equivalent of "save for half", but extended to martial attacks too.
Some wizard encounter powers similarly deal half damage on a miss, but for most other classes it is confined to dailies.
Fighters also have an at-will power - Reaping Strike - that deals STR bonus damage on a miss. And there is a feat that has a CON minimum, and if you take the feat then your attacks with maces and hammes that otherwise would deal no damage on a miss deal CON bonus damage on a miss. These are probably the closest abilities to the 5e GWF DoaM.
Another feature of 4e is that minions never take damage on a miss - so you can't get guaranteed kills against cannon fodder with your DoaM abiliies. There does not seem to be a similar rule in 5e, which would mean that DoaM fighters are quite strong, in 5e, against cannon-fodder opponents like goblins and kobolds.
I haven't come across the suggestion that 4e DoaM is abusive, and certainly haven't encountered it in my own game, but I don't hang out on 4e CharOps boards.