The sockets on most cars are rated as 15 amps, sometimes 20 amp. A few accessories such as some electric tyre pumps, and electric coolbox can draw over 10 amps each so if running more than one accessory at the same time using a socket splitter you need to ensure the total does not exceed 15 amps .Whats more, cheaply made splitter devices may not even be able to carry 15 amps, and the device itself may melt or catch fire at a lower amperage. Despite being fused.
I had this happen to me once when ,without thinking, ran an electric coolbox through a cheap multiple splitter. The wire lead to the splitter device overheated and melted the plug. (The fuse didnt blow). I was lucky, the device was the weakest link,and failed at only about 10 amps IIRC. In the end I fitted a second 12 volt socket wired to an empty slot on the fuse box rather than using splitter devices. But i wouldnt want to mess about adding a second socket to the Jazz, certainly not while its under warranty. Had the splitter allowed me to exceed 15amps it could have been the car wiring that overheated.
So the dealer was right to point out the potential dangers of a splitter. Phone chargers and the like are not normally a problem as they only draw about 2 amps each.
But I dont see how the splitter can be blamed for problems with other functions on the car, which I would have thought have their own wiring circuits. The socket should be able to provide 15 amps without depriving other circuits of the power they need. If not then the car is not fit for purpose and Honda need to rectify it - not tell owners to stop using things. Whether it be revising the wiring circuits , or fitting a bigger 12v battery.