I'll go ahead and have a stab at trying to identify what people might mean by "console games on handhelds":
1.
Straight ports of console games which weren't designed specifically for handheld - this in my view is rather inane as even the handheld most notorious for "console-like" games actually had very few (if any) direct ports of console games (excluding BC ps1 games, as these games were never designed for the PSP and are emulated on the device thus are technically not "PSP games" and shouldn't be used in the argument "PSP has console games".
2. Games not suited to handheld gaming due to control interface limitations - this one is pretty legit in my view, and given that the PSP had a few games like this, which were game genres & types traditionally thought to be better suited to "twin stick" control as with a console controller (e.g. FPS games etc)... however this would be a none issue with NGP ;-)
3. Games with big console-like graphics and production values - again this is a rather silly view taken by a few i would suspect have never actually played some of these games. Some of the best and highest quality titles in handheld gaming full stop, exist on the DS and PSP (e.g. KZ:Liberation, TWEWY, MGS5
W, VC:2 & 3, Patapon 1&2, Locoroco, Pokemon games, FFVII Crisis Core, GOW: COO&GOS , GTA:Chinatown Wars etc etc). These games are the pinnacle of handheld gaming and are 100% built & designed for handhelds.
Someone else can help me out here as i can't think of any more possible definitions. Even the three that i listed are none issues in my view with both the current DS and PSP. Both console's game libraries are diverse enough to cover every aspect of dedicated handheld gaming. The majority of the negativity that surrounds this notion of "console-like games on handhelds" is based on more presumption, misinformation, lack of any kind of extensive experience with the current-gen handhelds and just plain falacy in my view. There's not one single argument against deep, creative, mutlifeatured, long-lifespan, and complex (all aspects of console-like games) that can stand up to close scrutiny.
Even most of the casual iOS games, the one-minute, pick-up and play, shallow, two minutes of fun games are more often deemed as shitty as the shovelware that devs and pubs of old used to try flogging on consoles before the market ultimately rejected them (to varying degrees each gen). The games that do well on the iOS platform, the ones that people get excited about are deep enough and rich enough in terms of presentation, content and depth that they wouldn't appear out of place at all on XBLA & PSN. So please tell me again, what are these fabled "portable" games that don't exist currently on the DS and PSP that do so in abundance on the iOS platform?