Strange outlet behavior
This feels like the washing machine has a ground fault, where there's a connection between neutral and earth somewhere. See Why does my laundry machine trip the GFCI when I plug it in? 

If so, this is something you need to deal with, as it makes using your washing machine unsafe.

(In that case, when the tester checks for ground, power goes into the earth pin, up the earth pin on the washer, out the neutral pin on the washer, and back into the panel, which looks a lot like a real ground.)

You can also test with a different three-pin device connected, and see if you still get a fake earth reading.

EDIT: since you tested with a GFCI, and it works, I'll take another look with the new information.

When you plug in the washer, the 3rd light comes on. The usual meaning for the third light is that there's a voltage difference between hot to ground, which means the ground is going somewhere. While it's possible there's a grounding wire somewhere that you missed, another interesting possibility is that the surface the washing machine rests on is acting as a ground. This is somewhat dependent on the surface and the actual construction of the washing machine, but if the washer feet are metal, and it's resting on concrete, there might be enough conductivity to give a ground reading. In that case, your tester is giving a correct result. (But you should still run a real earth wire.)