This is normal for mobile homes
Mobile homes that have the label you see on them are designed to be fed by an external service pedestal that contains the main disconnecting means. As a result, the service pedestal outside the mobile home contains the neutral-ground bond for the attached mobile home's electrical system, with the panel inside the mobile home wired as a subpanel, as per NEC 550.32(A):
The mobile home service equipment shall be located adjacent to the mobile home and not mounted in or on the mobile home. The service equipment shall be located in sight from and not more than 9.0 m (30 ft) from the exterior wall of the mobile home it serves. The service equipment shall be permitted to be located elsewhere on the premises, if a disconnecting means suitable for use as service equipment is located within sight from and not more than 9.0 m (30 ft) from the exterior wall of the mobile home it serves and is rated not less than that required for service equipment in accordance with 550.32(C). Grounding at the disconnecting means shall be in accordance with 250.32.
Your shed setup sounds fine
The setup for the shed you described, though, is fine -- there is nothing that prohibits a subpanel from feeding more subpanels, as long as neutral and ground are kept separate through the sub-feeder, which you did from the sounds of things. Just make sure the shed has its own grounding rods and the bonding screw in the shed's panel has been removed!