When Alibaba got Amon from a Dungeon, his sword became a Metal Vessel, because Amon moved inside Alibaba's sword after he completed the dungeon. In other words, Alibaba became a Dungeon Capturer.
Dungeon Capturer (迷宮攻略者, Danjon Kōryakusha) is a person who
“completes” or “clears” Dungeons. If they survive the Dungeon, most of
the time that person acquires wealth and magical items, like a flying
cloth or Djinn Metal Vessel.
But Morgiana has a Household Vessel. Because Morgiana isn't a Dungeon Capturer she is a Household Member.
Household Members are people who help a King Vessel capture Dungeons
and forms a Household Vessel from that person's Djinn, or if they form
a strong connection to the King Vessel. If the King Vessel does not
have his/her Metal Vessel, the assistant can not use their Household
So, Morgiana was at the Dungeon when Alibaba got Amon and his Metal Vessel, then Morgiana made a connection to Alibaba's Amon. After this connection, was necessary an activation, and this only happened when they were at the dungeon of Zagan.
Here is the list of all Dungeon Capturer and their respective Household Members.