While dog licenses were common in other countries (and still exist in some), they were never implemented in the UK.
It's possible you're thinking of the Dog Licence which was in place in the UK from 1878 to 1930. However, this was not a tax on dog ownership itself, but rather a license required for any dog over six months old. This was similar to the concept of a vehicle registration, ensuring dogs were identified and controlled.
In short: Owning a dog has never been directly taxed in the UK. While there were licenses for dogs, they were not taxes but rather registration requirements.