Because a dog licence wouldn't be paid by the thugs and irresponsible dog owners. They don't even pay for car tax and insurance, so why would they bother with paying for a dog licence.
With the Swiss system, if you see someone with a dog, the owner must have their "permit to own a dog" with them, same as carrying a driving licence, and dog wears a special tag.
Who would police it? Authorised dog trainers, rescue centres, RSPCA, KC registered breeders, dog wardens etc. theoretically this would lessen their work burden because there would be less dogs in rescue and less dogs dumped.
Pet shops banned from selling and dealing in dogs. The puppies and dogs come from puppy farms either in the UK or Ireland. These dogs aren't socialised which is a potentially bad start for an animal and ends up with behavioural problems. including agression.
Allow members of the public to report dog owners that don't have the necessary permit to own a dog(s).
Microchipping or tattooing to be compulsory, and for the chip to be registered to the dog's owner. The number of stray dogs at the moment that are chipped but chip not registered is unacceptable.