By default, the Alliance application allows usage of the same serial number for different items but only if the items are different products.
If you receive a warning the serial number already exists and are sure there are no other items of that product having the requested serial number (use Item Locator to search for the serial number) then check within the Logistics Settings if the option for Unique Serial Number is enabled.
If this is enabled, then the validation for serial numbers is looking across ALL products and not just the one you are trying to define. Default value for this is disabled.
Same goes for the Unique Tag Number.