Personally, on the one character I brought over from CoH I just leave it vague - she's from an unspecified alternate dimension in which she lived in an unnamed city full of numerous superheroes, and is unable to get back for unspecified reasons. The only time I get into details is when I meet someone else from CoH or a champions native asks about it. Works well enough for me.
That would seem to work rather well in my eyes- it isn't overtly pushing "the other game" for CO natives, but it preserves your backstory. There have been plenty of other examples of "alternate dimension" backstories, so what's the harm?
It also helps avoid one of the other issues- copyright. Starsman alluded to copyright issues, and they do exist. Fansites like the PRIMUS database generally can legitimately reference and derive works off copyrighted material of the game they're promoting, but (for example) if NCSoft asked PRIMUS to remove references to their properties ("carnival of shadows" showing up in players' backstories there) they'd probably be forced to do so. They don't actively hunt down and expel their users, but by listing it as a prohibition, they're covering their butts should someone come a-calling.
