Hello!
This is not a bug, its a feature. As stated in the FAQ of the page, each player may only have one entry. In your case, the player instance on the local network is different from the one on Unity play. You may be asking why this happens.
I explained why this happens in one of my devlogs in itch.io:
Many people were frustrated about why a single player could have only one entry. Well that is because each player will have a unique identifier attached to them on initialization and it will be stored permanently in PlayerPrefs. With this, players can edit or delete only their own entry. This identifier is also used to determine which entry that is being displayed on the leaderboard belongs to the player. Several people apparently were not satisfied with this, so I made the LeaderboardCreator.ResetPlayer() function. After uploading a new entry, you can call it and the next time the player submits again, instead of overwriting the previous one it should create a new one. Just note that the previous ones would be unable to be changed or deleted afterwards.