In the UK, there are approximately 5.3 million dog-owning households (21% of all households) which combined own around 7 million dogs. The average dog-owning household owns about 1.3 dogs.
In the US, there are approximately 48 million dog-owning households (38% of households), owning around 72 million dogs. The average dog-owning household owns about 1.5 dogs.
To put it another way, in the U.S. we acquire over 7 million new dogs a year in order to replace those dogs that die due to disease, accident, or age -- more dogs than are owned by ALL people in the U.K.
To put it still another way, the U.S. has about 5 times the population of the U.K., but we have about 10 times more dogs because we have more households per capita, a far higher percentage of households owning dogs, and each household also is more likely to own multiple dogs.
These numbers are approximate. In the U.S. some sources put the total number of dogs at over 77 million and the average number of dogs per household at 1.7, while there is a similar "wobble" in the data on the U.K side.