I noticed that other tools (I tested rapier, hatchet, lumber axe) than pickaxes doesn't level up. Only players own skill with them levels up. I think that the other tools should level up too.
I think he is talking about the mining level of tools, rather than the skill level which gives the bonus modifiers. To answer what I assume to be the request, the only reason the pickaxes have a mining level increase is that they start one level below their normal level. What is referred to as the player's skill level is what gives the tools bonus modifiers.
I think there are some shovel blocks that require a mining level slightly higher than stone, although tools other than picks default to the higher mining level, IIRC.