The Sakai Katana is Jin's main sword and the Sakai clan heirloom he carries all game. It is your only real melee weapon besides the tanto, so upgrading it at a Swordsmith is the main way to raise your sword damage.
Every stance, parry and Mythic sword technique runs through the katana. Each upgrade raises the damage it deals, so you kill faster. Stagger damage is set by your stance, not the katana level. You can't pick up other melee weapons like spears or shields from enemies.
Level III: 300 Supplies, 25 Iron, 5 Steel. Level IV: 450 Supplies, 40 Iron, 10 Steel, 1 Gold. Level V: 600 Supplies, 65 Iron, 20 Steel, 2 Gold. Level VI: 750 Supplies, 80 Iron, 40 Steel, 4 Gold.
Swordsmiths work at Golden Temple and Komatsu Forge in Izuhara; Akashima Village, Yarikawa Stronghold and Umugi Cove in Toyotama; Jogaku Temple in Kamiagata; and Fune's Refuge on Iki.