VEVOR 50W 12V Solar Battery Trickle Charger Maintainer, Mono Solar Panel Battery Charger with Upgraded MPPT Controller & Adjustable Mount Bracket, IP65 Waterproof for Car Boat RV Trailer House Roof

This product comes with only one set of alligator clips, so it is designed to maintain and charge one 12V battery at a time; you would need additional equipment, such as a battery isolator or a parallel connection setup, to safely charge two batteries simultaneously.

No. You do not need to disconnect it while the engine or alternator is running. The included MPPT charge controller is designed to regulate charging and can be connected while the battery is also being charged by the vehicle's alternator.