Kirby isn't like Puff. Puff's best traits are her aerial mobility, safety, high range, and OHKOs. Kirby's good traits are almost all ground-related: his duck, grab range, and tilts. You can toss his Bair in there, too, I guess. His air game on the whole is very lacking, however. Consequently they don't play similarly at all.
Pichu plays like Fox in that they employ similar strategies with how they place their Nairs and acquire first hits. This is mostly as a result of their good movement games and having a similar speed class in their attacks. The difference is that Fox is a bit faster, has a bit more priority (so some moves Fox can muscle through, Pichu has to avoid, block, etc.), has a lot more safety on hit (Shine), and transitions from his first hit game into his punishment game far more effectively.
edit:
I personally find Kirby far more comparable to Sheik, quite honestly.