The arguments put forward by United fans on here as to why Liverpool, and not City, are United's "real" rivals are fundamentally ignoring/misrepresenting/misunderstanding what the term "rivals" actually means, from a true footballing perspective. They've manipulated rivals to mean "the team at the time who you are directly challenging for trophies". Well, yes, that's A type of rival, but clearly it's not the type of "rival" we're talking about here. Our rival this season is most likely to be Chelsea (poor start not withstanding), however we wouldn't begin to compare the rivalry between City and Chelsea with that which exists between City and United. If Chelsea are our closest rival, on the pitch, for the next decade then sure the "rivalry" will likely intensify and become more important to us, but again, it still won't match that which exists between the two Manchester clubs.
The reason many view Liverpool as United's real rival, ahead of City, is down to modern football and how it has changed the game, and those who watch it. English football has become more inclusive, the amount of fans a club can claim are "local" to the side has, as a percentage, decreased exponentially over the last few decades. If you were to work out how many United fans were from Greater Manchester in the 1960's, and then try and establish the percentage form 2015 then you'd discover that now it's a substantially smaller value. The same is true for all clubs, although United's supporter base is likely more diverse than many. Fans who are not "local", who have never been to Manchester, and have no affinity with the City, just a liking for the football team, will not view "rivalry" in the same way. They don't care about the history, they don't care about local bragging rights, they care about who can stop them winning things. They also care about what the media, which too has changed the face of modern football, tells them to care about. SKY, and other media outlets, have pushed the United v Liverpool "rivalry" for their own benefit, they call it a derby when it categorically is not one. It's a rivalry that has been bolstered, it's a rivalry that has been artificially supported and advertised. Of course there was a degree of rivalry before these modern influences came to bear, in the same way as there was a degree of rivalry in the hooligan filled decades of the 70's and 80's between many sides competing within football. United's quest to "catch" Liverpool's trophy haul will also have added to this, regardless of outside influence.
No-one is trying to claim that there isn't a rivalry between United and Liverpool, however it would be idiotic to try and claim that the current rivalry that exists has always been there, that it was the same in the 1950's, or the 1960's. In footballing terms the United v Liverpool rivalry is a "modern phenomenon", if it hadn't been for Ferguson constantly stoking the fires, the media buying into that for their own purposes (having a rivalry between the likes of United and Liverpool can be nothing but beneficial for TV companies when it comes to interest in football), and the makeup of your average "football fan" changing fundamentally over the last couple of decades then this rivalry would have waned with Liverpool's drop from footballing relevance. The reasons the rivalry is still perceived to be fierce, despite Liverpool being no real threat for at least a decade, and probably longer, is because the rivalry serves a purpose, a financial purpose, to many groups. If you tell someone something often enough they eventually assume it must be true, they buy into it, and they find their own thoughts on the matter are infected by it.