The Industrial Revolution in the UK started about 1760 and ended some time between 1820 and 1840. It was an era of the rapid development of industry due to the introduction of machinery. The machines were powered by steam. Factories sprung up and the mass production of consumer goods began.
This era made many dogs either indirectly or directly surplus to requirements because their tasks were no longer available. The world had changed and the role of dogs as working dogs, as I understand it, evolved into a new role, that of a companion. Also, because of an improvement in animal welfare and a greater sensibility towards companion animals and their needs, legislation was introduced to outlaw cruel sports incorporating dogs such as bull-baiting, badger-baiting and dog-fighting.
Dr Desmond Morris argues that at this time competitions for best dogs were set up in public houses in the 18th-century as a reaction to find some other use or entertainment from their dogs. These evolved by the 19th century into organised dog shows. The first dog show organised by Crufts took place in 1886. Breed standards were drawn up and enforced via competiton i.e. dog shows with the winner meeting the breed standard the best. Dog breeding became popular and the Royal Family participated.
In 2020, the American Kennel Club recognises 202 dog breeds. The British Kennel Club recognises 211 dog breeds and the Fédération Cynologique Internationale recognises an astonishing 344 dog breeds officially. This is far more than the number of cat breeds. Many years ago I did a video containing 104 cat breeds. This is far more than the usual number.
The point that I’m making is that there are far more dog breeds then cat breeds in Western countries due to breeding dogs informally (before the dog shows and breed standards) over a longer period of time than cats. This in turn is because dogs were domesticated before cats by, arguably, thousands of years. And once again the reason for this is that dogs were more useful on a utilitarian basis than cats so they became working companions to people much earlier than cats.
The theory about the industrial revolution belongs to Dr Desmond Morris. It is an interesting one. I have added my thoughts, perhaps unwisely.