It has its own internal battery, but may either use power from the key switch to start counting hours or possibly use engine spark impulses to trigger the clock. Normally the change oil reminder will auto reset after 2 hours of run time.
Most mower do not need to the seat switch to be activated start the engine, only when the pto is engaged or lever/parking brake is deactivated does the seat switch come into play and will kill the engine if you are not on the seat. So most likely you have an issue with the pto switch, brake switch, or a direction control lever safety switch. or if present could be a problem with an associated relay.