Small cot-sized beds are often found in pairs: in barracks, dormitories, prisons, hospitals, children's rooms, and other places. Perhaps such beds were frequently referred to as "twin beds" formally or informally at the time that bed sizes were being standardized, and furniture manufacturers just decided to keep the name that everyone had been using for that size of bed anyway.

So why are they called "twin beds" even when they don't appear in pairs? It may be something of a backformation: if two such beds are twin beds, obviously one such bed must be a twin bed.

Double or Full is larger than a twin (single) and smaller than a queen. Double means two people (double) can sleep on it if need be and the term FULL I think is because it is a full sized bed, not a kids single bed, or twin.... lol 2 twin beds side by side IS a King Sized bed. I think the term TWIN refers to the variations in which it can be used... it has a "twin" to make a King, a "twin" to make bunk beds, so if a child is part of a twin set..... Identical, etc., THAT person would refer to themselves as a twin.

In the 1950s, a single was as wide as the average person's shoulders. A twin was a single & a half. A double was two singles. A queen was a single & a twin. A king was 2 twins (or 3 singles). Nowadays, the single is forgotten, so everyone is confused. When I was a child in the 1950's, my bed was a twin and you could put two first graders in it side by side. My best friend had a single bed, so when I slept over at her house, I slept in her bed and she slept on the floor.

This a layman's addition to RiMMER's answer.

Think of twin shorthand for twinned twins and double shorthand for double-sized.

Double beds are used for partnering bed space (hence their usage for romantic couples, close siblings, owners and their pets, etc)

Twin beds are used for increasing living-space density. Traditionally, twinned twins are 2 beds separated by a walk-space in the same room, but they have a generic definition which is:

Twinned twin beds are 2 beds using the same living space. [see RiMMER's answer]

This living-space factor has made twin beds not only popular, but useful in prisons, dormitories, and barracks, and it is the standard bed size for bunk beds.

TL; DR:

Think of it this way, you don't give a married couple twinned twin mattresses, and you don't give prisoners double-sized mattresses (dropped soap-bars would be a problem)

I sell beds in the UK, and the sizes that we have are small single bed which is 2' 6" wide and 6' 3" long; we have large singles which are 3 feet wide and 6'3" long; and these beds can all be joined together with what's called zip and link, where the bases bolt together and the mattresses zip together. We have a small double, 4 ft wide and the same length as singles; a double bed which is 4' 6" wide, same length; a king size bed which is 5ft wide and 6' 6" long; and a super king which is 2 x 3' 6" beds that zip and link together, but you have to have custom made bedding fit the super king. That's the UK system.