Electric current must flow round a "circuit". If the inverter is connected to the battery by one cable, then that is the cable the current flows through. If the cigar socket is totally independent, then there is no reason for current to flow through that.
: if you joined one wire to another wire, eg to run an extra rear light off the original rear light, then the original cable would have to carry extra current eg original lamp 5A, extra lamp 3A there would be 8A flowing in the original cable upto the point where you have placed the join. This is the way that some cables get overloaded.
[ It is far more likely to happen in your home, where you may use a 13A extension lead to power a 10A kettle, a 5A TV and a 1A reading lamp. You could have each appliance on by itself without any problem, or the TV + lamp (5+1 = 6) or the kettle + lamp (10 + 1 = 11) but if you put the kettle AND TV on (10 + 5 = 15) then you would be asking for trouble. Theoretically, the extension lead would have a 13A fuse, and this would "blow", cutting off the current. Theoretically, the extension lead wires could cope with more than 13A "just in case". Worst case scenario: the fuse passes 15A but the wire overheats, melts the insulation, catches on fire and the house burns down, killing you, your family and all your guests