I think the intent of the OP is to install both a fused spur (feeding something, but limited to 13A) and also install another unfused spur, from the same double socket, to a single socket.
Now this is nominally permissible, there are no regulations limiting where a spur may be taken from, though there is a recommendation that the number of spurs does not exceed the number of points on the ring and also that the loads should be distributed about the ring evenly and even further that the length of a spur does not exceed 1/8th of the length of the ring (puts paid to all those distant garages spurred off the house ring!).
However fitting four cables into a double socket is always fun to do and hard to get secure, but as far as I am aware there is no specific limitation to stop this.
I think the information about an unfused spur says it can feed one point, this is usually a double or single socket but a quad socket could also be installed as it would be independently fused so effectively four sockets on a fused spur with the fuse at the end of the circuit and the 2.5mm² cable protected against overcurrent by a limited load (nominally).