You must play meaningful games against one another to enter into the discussion of a rivalry.
Totally disagree. There are really 3 components to rivalry.
The first is that the teams are competing closely with each other for some goal or other. Everyone is out to win the SB, or any sports' championship. But there's an added level of gunning for a division/conference/league title, or even 2 teams that are frequently in direct competition for a wild card as the season winds down. This aspect of competition becomes more intense if the 2 teams get to play each other because a win means twice as much - it helps you and hurts your competition.
The second is that the teams are both relevant in the race for the given goal. It's hard to say that 2 teams in a division are rivals if one is projected to be 14-2 and the other 2-14.
The third is that the teams have satisfied the first 2 requirements often, especially in recent history.
The point of all this is that 2 teams in the same city, aside for competing for their division/league title etc, are also always competing to be "the team" of that city. This is more true in NYC than anywhere else. The bigger issue is that so often the 2 NY sports teams in a given sport aren't frequently competitive at the same time. So they lack the history aspect. But in a year where they're both in the playoff conversation (like this year) or they play each other (like this year), the rivalry aspect is in full force. And anyone who followed the Jets and Giants in the weeks leading up to their game knows that the teams and their fans felt that way.