That's the thing, they are all rated for 13amp but it's not a constant load rating. Same as the plug in 110v transformers they are rated for one figure but the constant load is not far off half the headline figure.
I'm not sparky and it's not really an area of expertise and I think the household sockets are tested to something like a 20a max load but If you look at the big ticket 13a items like kettles, washing machines, ovens etc they all pull somewhere between 2-3kw but they only do it for short spells whereas an EV granny charger is pulling 10 amps or 2.3kw-2.4kw for hours at a time or at least they shouldn't be pulling more than 10 amps have read some of the cheaper non OEM ones of doing more.
There is specific ev rated single sockets, they have bs1363 EV stamped on them and are rated to pull high current for hours on end. Not easy to find though as no one seems to list what they are on any of the sites, only information seems to be the odd outdoor socket saying not suitable for EV in small print, I had to buy them through "tough leads" and turns out they are a hager socket
Have seen quite a few pics of melted sockets although it could be a number of factors causing it, loose wire, old cheap socket or simply just a standard 1.25mm2 extension cable plugged in