The origin of trainers must go back over a hundred years to the early 1900s, when the vulcanising of rubber made possible the large scale application of durable soft rubber in footwear. In the US, they were called sneakers and are still called the same today. In different parts of the world they are called sports shoes, running shoes, and many other things.
The possible source for the name trainers is thought to come from 1968, when the manufacturer Gola had a design called trainers. The name trainers probably comes from the shortening of training shoes.
One of the most popular brands, Adidas, first saw the light of day in 1920, under the name Dassler, the surname of the founder, Adolf Dassler, in Germany. They were used in the 1928 Olympic Games in Amsterdam.
Whatever we want to call them, and in the UK at least, trainers is still what we want to call them, trainers are the most popular casual shoes in use today and ladies love trainers.