I only know very broad strokes like most kennel clubs were formed in that time and that's when most breeds first became distinguished, but what were the reasons for it? Was it a general cultural phenomenon or just a dedicated group of hobbists? Anything at all on the subject would be helpful.
Once kennel clubs were established, the breeds and the breed standards were differentiated from each other. Naturally, as each breed became its own specialized dog, tensions began to rise. Two different factions of dog fanciers arose: the gentlemen breeders (and they were largely men until the late 19th century) and dog dealers. The gentlemen were from wealthy families and felt they knew more about breeding due to their own backgrounds. While the betterment of the breed was their stated goal, it pretty clear that that meant different things to different people.
The first dog show was held in 1859 in conjunction with a cattle show. The only breeds shown were hunting and sporting dogs - a favorite of wealthy upper class gentlemen. Other breeds and groups would be worked in as the century progressed and as more people of varying classes became interested in preserving their breed of favor. Women would become more involved as smaller and toy breeds became popular. One particular woman who heavily influenced this trend was Queen Victoria herself. She favored the Pomeranian - a spitz breed. During her lifetime, the breed was reduced in size by half (and is even smaller now) than its progenitors. She also favored the iconic red color over their traditional white.
Queen Victoria was highly influential in terms of how pets - in particular dogs - were treated. Before her, dogs were largely tied out or left to their own devices unless needed like a tool. She showed they could become a part of the family.
Around the time that dog shows really started to build up steam, the middle class was forming and growing. Breeding dogs became seen as a hobby primarily for wealthy folks. This was because they would have more leisure time than the working class. At that time, dog shows didn’t just display the best looking dogs, but were a way for people to show off their own pedigree and wealth. In fact, many shows were split in two, with the working class dog breeders having their own prizes and all. This would eventually change, just like the exclusion of women from show, largely due to the influence of the USA and popularized dog fancy.
One thing that did stand out was the creation of new or ancient breeds - the most widely known example to be the Irish Wolfhound, an ancient breed that was extinct long before the Victorian fascination with dog breeding. One man, George Augustus Graham, decided to bring the breed back through various breedings of different types. He was condemned by many in the hobby for his breeding practices, though he did eventually create a new Wolfhound.
So all in all, the rise of the dog fancy coincided with the rise of the middle class, prompted by Queen Victoria. While it had started as a way of distinguishing wealth initially and showing off a gentleman’s own pedigree, as they middle class rose, so did the proliferation of dog breeds.