Nezuko is initially 12 years old in the manga. As the story unfolds, her age doesn't change significantly in a traditional sense, but her character development is quite remarkable.
In some fanfictions, he might find it in an ancient weapon stash hidden in a mythical location. Maybe while on a quest, he stumbles upon a secret chamber where the katana is kept as a powerful relic from an old civilization.
A simple Nezuko and Zenitsu fanfiction could be that they go on a journey together. Along the way, they encounter other demon slayers who are initially skeptical of Nezuko. But Zenitsu stands up for her, explaining that she's different. This shows the depth of their friendship and how Zenitsu has come to trust Nezuko completely.