That is a beginning. The setting made me think of a japanese game first. Maybe you played a translated game? Is the art more westernised (think cartoon, as you mentioned) or more eastern style (think anime and manga)? We should exclude or include this possibility. I recommend to take a look at the
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for japanese titles or
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in general maybe you find the game per chance.
Also do you remember where you found it? Newgrounds, diverse download websites like sxshentai, svscomics, some forum like Hongfire, Undertow Club, Fenoxo's, or Legend of Krystal maybe?? A japanese download source would be e.g. DLsite (again, your setting sounds so jap-like to me). The more you still remember the better - any names of npcs, of locations (other dimension got a name? Name of school?), of items, also special clothes.
Anything you can recount.
What could help too is where and by whom it could have been created as I wrote. You 'working as a janitor' sounds like an Illusion title (Immoral ward) to me, but they use 3D environments. I don't really know of a western game atm with such setting, except for some russki games which can have weird plots sometimes (including magic a.o.). Remembering the 'look and feel' of the platform can help, they all make you play a game in a
specific way. Did you play another game where the gameplay felt very similar (mechanics, stats, typical elements like buttons and such)?
- You can check the QSP, Adrift and RAGS game threads we have here, especially the OPs with screen caps. Maybe you recognise one of those engines? They do not offer a 2D environment, are more AIF style (you click commands in a 'player app' and got pictures to show you npcs and progress). But some of them got a pretty fantastic story,
- Check RPGM threads we have here, especially the screen caps. Look at the OPs here or do a web search, confirm or deny this could be the platform. Would be nice to know if it could be a RPGM game, as this would help keep focus on this platform (it sounds like that or a similar platform is used). Games with RPG Maker engine have a typical style to them. You mostly got 'chibi' characters running around in a 2D environment, like in good old NES games and such (think Zelda). If a scene starts you get renders and even animations.
Last I can think of is good old Flash as a platform. Those games have typical 2D environments to move around on screen. But this environment you should know, it is displayed to you because games may start in 'flash player'. There are some eroge VNs we have here. Marble Syrup and Spiral Vortex are known to use this platform. SV e.g. has sci-fi settings, but mostly you play a chick doing things... Have not played all MS games. You got university setting in one, you play a guy, but I believe they do not contain the gameplay you described.
Report back. I like challenges. At least you are occupied for a while now.