To answer your title question, simply because it's easier that way. The decimal system operates on 10s, so we have decades, centuries, and millenia. When noticing change in history, whether it be from social, economic, political, or other causes, it is easy to point to a decade and say "The Sixties was when this changed." It works as a general framework for looking at periods of time, and because it's a simple counting system, it's easily adopted to general usage.

I believe that also answers your second question somewhat. In general, cultural changes don't fit neatly into periods like this. Also, in my opinion, there was an attempt to force nicknames on the later decades of the twentieth century, purely because of the earlier ones being so well known, and the later generations feeling a need for identity. However, I wouldn't let that necessarily diminish the view of those decades, as the Roaring Twenties and the Sixties were also key to identity. As for cultural changes and how they fit into the frame, each decade is approached differently.

The twenties are generally considered to be 1920-1929(and then there are the oddballs who suggest other measurements). The twenties get their nickname as the 'roaring' decade from the lifestyle that dominate images of this period. Parties, raucous behaviour, new fashions that challenged the old, and new forms of music all contributed to this. The ostentatious lifestyle was limited though. It wasn't available to the working classes, and the middle class wouldn't have felt the full effect either. It was a lifestyle reserved for the upper crust of American and European society. The roaring twenties don't completely fill the decade though. They're framed themselves by major events: at the start, the end of the First World War; and at the finish, the Great Depression. Because of this, the twenties are often observed to be a period that was somewhat romanticized by later years. It was a decade of wealth and peace, isolated between the worst economic disaster in the modern era, and the largest war the world had ever seen. It's easy to see how those who came after longed for the return of the glamour of the period from ~1919-29, and ended up referring to it with such an exciting nickname.

The sixties are quite different again. Using the same framework of decade measurement, you'd assume it was from 1960-69, and no more. But in reality the movements and culture that we associate with the sixties doesn't fit that well into such a tight constraint. Different historians have different opinions. There are hundreds of books written about the period, and they all have an opinion of the start and finish dates. Terry Anderson's book The Sixties covers the period from 1960-73. The December 1969 issue of Life magazine(the last issue of the numerical decade), said that the period could be seen in two parts, beginning with an "energetic shift from the calm of the late 1950s", suggesting a slightly earlier beginning. Arthur Marwick, in his book The Sixties(I know, I know. Historians have a knack for being unimaginative when it matters) opens with an introduction titled "Was There a Cultural Revolution c.1958-c.1974?" Eric Foner begins his chapter on the Sixties in The Story of American Freedom with the story of Rosa Parks, suggesting a start date in 1955. He finishes by noting that the sixties "overspilled its chronological boundaries", and that "It was in the early 1970s that movements born in the late sixties reached their peak of militancy", giving as examples the gay liberation movement and the American Indian Movement. While most of the key events of the Sixties happened within the bounds of 1960-69, many of them had their beginnings in the late fifties and ended in the early or mid seventies. Because of this, it's generally suggested that a more accurate name is "the Long Sixties", encompassing the entire period, from the rise of liberalism and the New Left through to the end of the Vietnam War.

Sources:

Eric Foner, The Story of American Freedom.

Terry H. Anderson, The Sixties.

Arthur Marwick, The Sixties.

The term "roaring Twenties" originally referred to the storms of those latitudes at sea. It was only applied to the decade in the Thirties.

Culture trends can almost be guaranteed to change in the middle of the decade during the late 1800s and the 1900s. Look at the bustles of the 1870s and 1880s: right from 1878 to 1882 they vanished. Shouldn't they have waited for 1880 and stayed away until 1890? They finally disappeared in 1905.

A big cultural break is 1912 (not 1910): tango, Titanic, & Tarzan. Other notables: 1934 transition from Hot Jazz Era to Swing Era, 1954 from Cool Jazz to Rock and Roll Era, 1964 Rock and Roll to Rock Era.

Personally, I loathe decadization, other than when very casually used. No one ever stopped and changed their ideas or fashions because a zero came up on the calendar. We didn't even for 2000 with the millenium flip. If ever we were going to start wearing skullcaps with antennas, you'd have thought we'd have done it then.

It's cheap, it's easy, it's fast, and it results in a lot of crappy history. Some writers/editors act like it's a law instead of a handwave shortcut.

4x, I once threw out a book on "The Forties" that claimed the New Look of 1947 was the female fashion for the entire decade. Like wartime rationing would allow women to wear ankle-length full skirts with multiple crins holding them up.