Doesn't he stomp on mushroom-men and kick tortoises? That's pretty violent.
And, while it is a great game, I wouldn't call it a great story.
The whole story can be summed up with "evil dragon captures the princess. Mustachio'ed hero braves danger to save her. Plot twist: our princess is in another castle".
If you want story get the Paper Mario games - they have a much more detailed story line (even if it’s still relatively simple).
Also probably not a good idea to encourage kids to eat any strange mushrooms they find.
You don’t have to kill anything to advance, I try not to considering how harmless the chestnut men and how friendly the turtles become in later games