Personally, I wouldn’t necessarily say it fits anyone in the DCU, or even in related Vertigo or other material, including perhaps the most obscure, save perhaps something I may not know well like Wildstorm. [For one thing, it’s a very repetitive song, for as much as it has great hooks, features Isaak’s vocals and interpretation well, but barely has any depth nor certainly any extensive amount of lyrical meaning].
It clearly has connotations of either madness or mental issues (and I’m not meaning that in a bad way) and desire and romance; while I personally never bought Lucifer and the detective’s relationship as the great romance the series sold it as, I also didn’t really buy Lucifer as a bog-issue police procedural wannabee fan, it seems really trite and boring. Wanting to punish evil doers is something the devil could do without all the paperwork and inter-office drama and other crap that comes with the LAPD, either in real life or this fantasy version of it where they aren’t overworked, messy and disorganized, and in typical police procedural television fashion, are investing all this time, money and other resources on solving just one crime.
No, except for both words in the title, Lucifer liking or wanting to play games with humanity, and Lucifer liking, wanting to encourage or wanting to be wicked itself, I would not ascribe the song to him, but perhaps to someone dealing with unrequited love, an obsessive love at that, after a heartbreak, which could well describe the late Golden Age Clark Kent, before the superd**** that came with Mort’s editorial intervention, and how he felt about Lois Lane.
A relationship that also, by the way, didn’t become a great and healthy relationship (e.g. no longer obsessive on Clark’s part, no longer obsessive on Lois’ part as to Superman, etc.) until well after the Golden and Silver Age, for as much fondness as I have for a lot of ways in which it was developed and even nurtured even in these late 1940s and 1950s stories.
If I wanted to play on the mental health aspects, I could ascribe it to certain denizens of a certain gallery of certain basic and boring villains a certain mentally unstable person who hits his head many times on the daily and yet still gets considered to be a great strategist and excessively intelligent, but I’d really rather not.