Thingwall markers are added to the player maps independent of character or even account. You can technically download and import a .hmap file, containing all Thingwall markers, before "discovering" a single Thingwall on any character.
Obviously character-specific maps aren't the solution, but the current system will inevitably lead to players asking themselves "has this specific character discovered this specific Thingwall?"
Currently, the only way to be sure is to visit Thingwall again or go to adjacent Thingwall (or chain of connected ones) and check if you can travel to it from there.
Previously tried to make this point here.
(This OP has been edited to communicate the issue better.)