What is this stinging/biting insect?
The stinger that got stuck in your partner's skin isn't completely black. Its coloration shifts from dark brown near the abdomen to black to dark brown again at the tip of the stinger.

This identifies the stinger as one belonging to Vespula vulgaris, the common wasp as the stinger of the other most common stinging insect in Scotland (Apis mellifera or European honey bee) is honey brown to brown.

Once in a while a wasp can happen to get its stinger torn of when it's flicked away during its sting. Obviously it dies afterwards as also part of its intestines are torn.

Pictures of both stingers can be found below.