This is a very good question and a lot more complex than it would seem.

It's important to remember that while the 1950's saw the birth of rock and roll, the entire country was not necessarily embracing it. On the contrary, the "stiffness" of the 1950's is part of what gave birth to the counter-culture elements that took hold in the 1960's. Bob Dylan notably cited this song as an example of the type of music that was popular when he was a child. There is an implied contrast with the type of music he would go on to play as a figure of the counter-culture, and "(How Much Is) That Doggy in the Window?".

Some of this is subjective, but "Doggy in the Window" is sort of symptomatic of where mainstream American culture was in the early 1950's. The decade is perceived as being a time of "innocence" for the country when in many ways, it's quite the opposite. Segregation battles were being fought in the south, the beatniks (a sort of anti-conformist precursor to the hippie movement) were stating to take form with the rise of poets like Jack Kerouac, and a war was happening in Korea. While this is happening, World War II veterans are starting families and America is entering what has been called the "Golden Age of Capitalism". A time that is still looked back on today with a certain level of naivety and longing.

The song captures the sort of "innocence" of 1950's white America in a nutshell. Patti Page, the singer of the song, was a performer who typified pop artists of the time. The song is simplistic in nature itself and hints at the growing American commercialization that is occurring all over the country. The dog is being seen as cute and something that a child would want, while at the same time being commodified and described as being on display in a window. It demonstrates where the attention of mainstream white America was while other events were taking place abroad and domestically.

It could also be argued that this song is in the vein of what I am terming "distractionary pop music" that still exists to this day. While the threat of nuclear war was starting to develop as the Cold War kicked in, this song contains lyrics such as this:

I read in the papers there are robbers

With flashlights that shine in the dark

My love needs a doggie to protect him

And scare them away with one bark

As an example of how this type of pop music permeates into future generations, take a look at the billboard charts today. A majority, if not all, of the music reflects the trends and fashions of the day with seldom any intentional reference to the political or social environment of the time.