The GoG version of Anno 1503 does not have multiplayer mode enabled by default.
You will need to do add the following to the registry to re-enable it.
HKEY_CURRENT_USER -> Software -> Max-Design -> Anno1503
Right click on Anno1503 -> New -> DWORD
Name the New DWORD into MULTIPLAYER and change the value into 1
Start Anno 1503 and enjoy the multiplayer