Problem: There are two characters with the same name on my server.
We both show up on the Lodestone search

I contacted the GM service. What I was told is that my character was first on the server, and the other character transferred afterward while having the same name.
In this case, a person is prompted to change their character's name before they can log in - but they don't have to. The character can remain with their duplicate name indefinitely.

This creates a problem for me personally. I was planning on Fantasia+Name change for Endwalker, but save the current name on one of my alts. I probably won't be able to do it since after changing my main's name, the "ghost" will retain ownership and it will become unavailable to me.

My Suggestion:
Check name availability before transferring your character. If the name is taken, still allow the user to transfer but let them know they'll have to change their name once the transfer is done.
The better solution would be to integrate the name change into the transfer itself if required. Something along the lines of, transfer all the data as-is other than their name.

By making the name-change part of the transfer, and preventing the transfer until such a change was made, the chances what happened to me will repeat is reduced.
At this point, I wouldn't be surprised if part of why the character hasn't been logged into (it's been there roughly since I created my character some five years ago) is because they had to change their name to do so, and chose not to