Deep character flaw
I don't know how they saw 'Jake', but it seems like at least some writers saw him as some kind of 'empty template' to try out all kinds of ideas. This lead to Jake never having a consistent character, besides the basic traits.
However, those traits betray the other things they tried to make him into.
For example, someone that's (admittedly!) lazy, bad at school, does as little as possible, eats as 'easy food' as possible (unhealth junk food), doesn't even brush his teeth (lost his toothbrush for 3 days and doesn't see it as a problem - why isn't he using electric tootbrush to begin with?), etc..
..can't be suddenly so deeply knowledgeable about some obscure Japanese manga, or manga character.
Jake is NOT a manga nerd, Jake is a lazy, fat simpleton that doesn't care about school, his own future, his own health, his body, etc. and he basically doesn't really even have hobbies besides just playing, eating and sleeping. And watching TV. Anything you suggest, he's gonna say 'TV'.
(Of course this shows bad parenting on both Judith's and Alan's part, but talking about that would deserve a massive thread of its own, it's pretty darn unbelievable that Supernanny exists and people are still clueless, but same goes about Dog Whisperer and bad dog handling you see every time you go out probably)
Furthermore, when someone like Jake - whose whole idea of everything is always 'lowest hanging fruit' - wants to play a videogame, he's gonna play the most basic, dumb, action-based simple games possible. He's not gonna read lore, he's not gonna familiarize himself with story, he's not gonna play a game that takes 20 hours to complete in one sitting (if you know exactly what you're doing and you edit out 10 hours of useless/random fights).
Jake is a Tekken/StreetFighter/Car games kind of guy. He's not a FINAL GAWD DAUMN FANTASY-kind of a kid!
Maybe there's some japanese enthusiast (the W-word comes to mind) among the writers, but they shouldn't vomit their obsession to the script so hard it damages Jake's character this bad.
There is _NO_ way Jake would a) know what Final Fantasy is b) want to play it c) be ABLE to play it (too much story, etc. - he couldn't even finish reading cliffnotes of Lord of the Flies!) d) have the patience to ACTUALLY complete the game, or get through even 5 per cent of it.
The whole 'I want the lates Final Fantasy' episode is just pure fantasy BS orchestrated by some writer that doesn't know or care about Jake, so he has this weird character flaw flux, where some day he's intelligent or interested in something completely out of character, and other days, he's dumb as a doornail, and only interested in food and bewbs.
They really don't think very far when they write these things, now, do they?