Welcome to Eorzea!
As Kacho mentioned, mediumship in Eorzea/Hydaelyn, is something not usually seen, and this is because of how the cycle of life and death works in the lore - all life and souls are born from the Lifestream/Aetherial Sea, and after a living being dies, their soul returns to the Lifestream to be reborn anew. Hence ghosts and spirits... usually either are just lingering aetheric traces that have been anchored to the physical plane, or, are not spirits at all (at least, not a spirit of the 'actual' world), but something else called a
voidsent, which is from another dimension called the Void.
What the Void and the voidsent really are, is a big spoiler that is revealed much later in the story (and plays an important part in it), so I will not spoil any more. But there is a Job class recently added called Reaper (RPR for short) that has as an ability where a very low level voidsent possesses the player's character temporarily, although the voidsent appears to be lacking in independent thought and is simply a construct of darkness-aspected ather. It is probably though the closest to what you're looking for, so might be an option for you once you're much further in the game (unfortunately RPR only is available at level 70, as that is the level it starts at). The quest to begin the path of RPR is found
here.