Location almost certainly plays a big part in it and which fans one comes into contact with most on a day to day basis. While it cannot be denied that the United fans I know in Manchester have a strong dislike of Liverpool, the derby has always had huge significance for them even if we weren't anywhere near United's level on the pitch. A mate of mine is quite clear as to which team he hates the most and his view pre-dates the 2008 takeover: "It's you blue c*nts because I have to put up with your shite, day in day out, whereas I don't know any Liverpool fans!" or words to that effect. Others aren't quite so certain and dislike both teams but are probably swayed one way or the other by who is the bigger threat at the time so when United and Liverpool were challenging each other in the mid-late 90's and City were doing our best yo-yo act and not even in the same division, the hatred was almost all directed at Liverpool whereas we were seen as more of an irritant for winding our red mates up in the pub when United suffered a rare defeat. And of course there's the general underlying theme of the Manchester/Liverpool rivalry. As such, City fans aren't keen on the Merseyside clubs either, although we've tended to have more problems with Everton fans down the years funnily enough.
Step outside Manchester and the immediate surrounding area though and it does indeed become a different story. There are more United and Liverpool fans in the country than any other so it's understandable that in most areas that two sets of fans will see each other as the biggest rivals. I'd imagine it's exactly the same in Ireland too. In London I think it's slightly different as there are other big clubs down there. I met a female United fan from London once and she despised Arsenal more than any other club - probably because she knew more Gooners than any other set of fans.
Tell you something though - during that 2013-14 title race many of us City fans felt the thought of Liverpool winning it would be every bit as unbearable as United winning it, if not more so, due to the huge media w*nkfest that would've ensued if Liverpool had been successful!