Metaphorically speaking, I've always thought of the king as an extension of you (the player), i.e. the strategist that gives the orders to the other pieces and only moves when it needs to. So I'd think chessmaster-like people like Yukari are pretty king-like, or maybe just leaders in general.
Not too sure about the rest. The queen is like the powerhouse of the game, so I guess someone who'd take an active role in fighting would be one of the protagonists, maybe. So Yukari would have Reimu as her queen? Which kind of makes sense. Although "Yukari's faction" isn't really large enough to fill the other pieces, I don't think.
It's also a little iffy since there are two rooks, knights, and bishops each but there obviously aren't enough characters that work thematically to fill each piece, but doubling up on the same character for both pieces could be weird too.