The Devil May cry games are thematically under the fantasy/dark fantasy (and, like you said, urban fantasy) type of video games. The themes do differ from each other over the games:
DMC 1 and 3: Classic dark fantasy theme (i.e. creepy gothic towers and castles), all in a contemporary setting
DMC 4: Medieval-ish/Renaissance-ish (based on Fortuna's architecture and design) contemporary dark fantasy (due to the interior fo Fortuna's castle, the hidden pit, etc.)
DMC 2: 90% urban fantasy, 50% science fantasy, 10% dark
DMC 5: Pure urban fantasy with a dash of science fantasy, almost like the Mortal Instruments, Buffy The Vampire Slayer, etc. This is pure contemporary/urban fantasy done right.
DmC: Devil May Cry: Tries WAAAAAAAAAAY too hard on the teen urban/science fantasy theming motif; littered with corporate images about obesity, soft drinks, etc. Steals too much from Buffy, Twilight, Futurama, Mortal Instruments, They Live, Bayonetta, etc. Itsuno and the DMC 5 team both did it better than Tameem and NT!
actually, DmC teen urban fantasy theme is only present in narrative and in the "real world".
the Limbo (the idiotic thing the actual level are set in) is a "videogame surreal" art style, something common in the 2000 (but nowhere near good as, for example, Evil Twin or shiny titles)
I actually hate DMC5... nearly the same as DmC...
the big difference is that when DmC was announced I hoped about a real DMC that felt "right". the DMC that was supposed to be that is a sludge of westernization that kinda looks like a western reboot.
eh...