Players have ideas about how training works, here are a couple:
If you set up a barracks and some dwarfs begin to spar, they will improve many fighting skills, but will not improve their Armor skill unless they are wearing armor in the location that was hit. So, the idea is to completely cover a dwarf in armor while they are training. It doesn't need to be light armor.
After a dwarf has developed good Dodge and Shield skills, they will improve their Armor skill slower because they are succeeding at dodging and blocking. So, the idea is to ensure your dwarfs work up their Armor skill before they are able to dodge well. This means you need to get them in armor when you start training them, and if you do not have enough metal, you put them in bone and leather armor.
If I am correctly recalling my last drillmaster dwarf, he started with 0 Dodge and 5 Armor skill, and trained in full armor, without shield or weapon, and become 15+ in Dodge about the time he hit 10 Armor skill. He never fought in real combat.
So, the important reason to train while wearing armor is to ensure their Armor skill benefits as much as possible. The armor doesn't need to be light, but it needs to cover most of the dwarf.
Because some players want to start training dwarfs during the first year and do not have enough metal to completely cover the dwarf, they make armor out of bone and leather.