Well, given we played them in a final on Sunday, and a semi-final last season, yes I'd say the two clubs are rivals.
Actually yes you are, but it's hardly your fault, nor is it surprising. For all the stuff about City being 35 years in the wilderness, had anyone told the City fans that's what was coming in the mid seventies, they'd have gone berserk. Everyone tailors their expectations to where they are in the pecking order, it's completely natural and normal. There was plenty of screaming on here last season with City coming third, if tenth place was on offer - despite it being a pretty decent result compared to most of the 80s, 90s and 00s - there'd be complete outrage, because we've got used to being at or near the top.
This is where Arsenal fans are now, and yes, while Hartlepool fans will say you don't know you're born, that's not really the point.