When the first Senators team moved to Minnesota, the fans didn't move. They stayed here and rooted for the second Washington team. And when that iteration left, the fans were still here, ready to root for the home team. Eventually some became sort-of fans of the Orioles, but didn't really have a team again until the Nationals arrived.
As a child, I rooted for Mickey Vernon and Bob Porterfield, Eddie Yost and Spec Shea. Now I root for Max Scherzer and Bryce Harper, Stephen Strasburg and Anthony Rendon. When they leave, I will root for their replacements.
When I think of the heroes of baseball who have made the most wonderful memories for fans in the area, it's not Tim Raines or Gary Carter. It's Frank Howard and Walter Johnson ... and Ryan Zimmerman (just names off the top of my head).
For continuity in the record books, I guess it makes sense to consider the Nationals as the continuation of the Expos franchise. But I doubt there are very many fans going to the Nationals games these days who first became fans of the team when it was north of the border.