Should sport games questions be mixed into one big category like nhl-series to prevent all those repetitive tags?
Much like for this question, no. Each game should be tagged individually. Tags are not expensive, and on Arqade they are not deleted when they're only applied to one question, so nothing is preventing us from tagging our questions properly.
In conjunction to what Wrigglenite has already mentioned, I want to highlight one other reason we should not be combining the games into a single tag. In your question you state that a question for nhl-14 might apply to a question for nhl-17 however, there is also a good chance it might not. The core mechanics may stay very similar, but there a number of things that change with each new game.
In the case where we only had one tag for the entire series, it would be much harder to search for answers specific to the specific game you are playing, whereas currently you can simply filter by tag. Tags are cheap, and help filter questions more easily, so we should use them for each game. There is no issue having a few duplicate questions across different games in a series.