I found that behind that 13A wall socket is a 20A circuit breaker which is shared with several ceiling lights and a washing machine. Does that mean the wiring behind the socket *should* take at least 20A, since the breaker is 20A?
Yes or No. The one who laid the wiring may put those good enough for 20A, or just good enough for 13A (which is cheaper and can cover most routine household use). You need an electrician to look into that. And since the circuit should be connected in parallel rather than in series, it's most likely that all the wiring from the circuit breaker is (up to) 13A only while the circuit breaker itself is good for 20A
(In case you are not familiar with parallel or series circuit stuff)
Think of the roads connected to the cross harbour tunnels, most of them are 2 - 3 lanes (some even are 1 lane only), but when you get close to the toll gates, you have many more lanes.
The roads connected to the tunnel are your wiring. The toll area is your circuit breaker. You may have many small roads (up to 13A) connected to the toll area (your 20A circuit breaker).
But why would charging at 10A using an existing socket with a 20A circuit breaker (albeit shared with a washer) cause a fire? Isn't that exactly what a circuit breaker prevents?
I guess you are referring to the comment from Mark about not running the washing machine at the same time. The Wall Connector and the washing machine use "two different sockets" in our eyes, but as jsju has pointed out, they may be sharing the same wiring behind those two sockets. You leave a good margin of safety running the Wall Connector at 10A, and the washing machine will take away that margin without you knowing it (if both your car and washing machines are drawing power at the same time)
If your Wall Connector and your washing machine are running on completely different wiring, let's say they run in opposite directions, it still could not eliminate the possibility of them sharing some common path before parting.
Let's put it in another way using the cross harbour tunnel analogy. Question is are the "Wall Connector road train" and "washing machine truck" using the same road? If so, since both are very big, big vehicles, you will have traffic jam (overheat) in that particular road (wiring) while the toll area has normal traffic (total current still under 20A). When you look at the traffic at the toll gates, we are having a beautiful day (circuit breaker won't be triggered). When you look at the traffic in one of the roads, we may be heading into a disaster.
Options are:
1) ask your electrician, he will know if the Wall Connector and washing machine are sharing any wiring or not at all; or
(As suggested above)
2) build a new road for the "Wall Connector road truck" (separate wiring); or
3) while the Wall Connector is in use, stop all other major user, i.e. the washing machine