Why would there be 2 lines into 1 socket?

[EFLImpudence]

Non stranded conductors I tend to twist together each (7 strands) of the each conductor together separately then, maybe, double accordingly too.

Of course.

 

[Taylortwocities]

@OM2 What are you going to do when (more likely when and not if) you put in a brand new socket and it still doesnt work?

Have you tried plugging something else in to the socket? Does that work?
Is there a fuse switch, perhaps above the worktop, that controls that socket? Or a grid switch array somewhere?
have any other sockets stopped working, or just this one?

PS. Those non-contact pens are great in some circumstances. But they only detect the presence of a live voltage. For your diswasher to work it needs a live and a neutral. That magic wand wont help you if the neutral line is broken somewhere, you need a two probe test device.

So it's important the wires stay connected into one socket?

Absolutely. If you separate them, either you have broken a ring circuit, and that could mean on overload of the cables. Or the next socket in the chain will not work.

The socket isn't designed to have 2 sets of wires per wiring (just my laymans uneductated opinion!).

Yes it absolutely is.