To me I just think the Ducks and Sharks, maybe not recently, but in the 2000s, when they played each other they elevated their game. It isn't a rivalry based on physicality, although there was plenty of hard hitting games, but just on pure skill and heart. And it is a mutual thing. I think people put too much stock into the playoffs. And the Red Wings, without the playoffs, they aren't rivals. If they never play another playoff game against each other that rivalry is nothing special, it would die. But the Ducks and Sharks already were a strong rivalry even before the playoffs, and will continue to be rivals no matter what. Even if one team sucks and the other team is a perennial division winner, that rivalry will still continue.
To me it seems like Detroit plays more different against the Ducks than the Sharks. So yes, I think the players do consider the Ducks a rival. Like mentioned before, Datsyuk has fights against the Ducks and only the Ducks.
But that has been going away a little since the Ducks have declined.
I think the playoffs are the biggest part of rivalries. Detroit & Colorado was the perfect example of that. Those two teams hated each other with a passion and it all stemmed from meeting in the playoffs.