Scores on signs do not display any score with "*" (wildcard) as selector.

The usage of the wildcard "*" selector is to show the player looking at the text their own name or score. It should always resolve to the player looking at it.
This works for books, before it resolves the text, it turns the * selector into the current player.
Also works for tellraw and title commands.

However in signs it doesn't work.

The command:

/setblock ~ ~ ~ oak_sign{Text2:'{"score":{"name":"*","objective":"my_obj"}}'}