In Europe and pretty much the rest of the world, the game is called football. In the US there's already the national sport, football, which the rest of the world calls "American football" hence the term, soccer, was adopted in the US.
The US national sport is (American) football
(see edit correction below)
American football as a whole is the most popular sport in the United States; professional football and college football are the most popular forms of the game, with the other major levels being high school and youth football.
But as we all know, it's a tough rough game. You need to be physically strong, heavy, and fast on your feet. It helps if you are over 1.90 m too. It's a game of strength, coordination, speed and brutal force; so it's fair to say that traditionally it is a man's game. It's one of the first sports that American dads would teach their sons, a sport where father and son(s) could bond.
As a result, moms and daughters could feel excluded, and if a son (for whatever reason) disliked or was terrible at football he could always turn to baseball or basketball. When soccer finally caught on in the US, it was initially played by girls. As testified by this article in the New York Times dated 1996.
By DONNA GREENE
Published: December 1, 1996
WHEN Deborah Slaner Larkin of Pelham, an advocate of sports for girls,
hears ''soccer mom,'' she cringes. The term, she says, does not do
justice to those who are making a difference in their daughters'
lives.
A member of the President's Council on Physical Fitness and Sports and
the mother of a 5-year-old, Ms. Larkin feels strongly about the
benefits of sports. She is on the board of the Westchester Fund for
Women and Girls [...]
We also are the people who are on the field. Most of us are not
coaches, although some are. I coach my daughter's soccer team. Other
moms are there in force watching. They see the condition of the
fields, they see how the coaches coach, how the other parents react.
They see how often their kids get to play. So they know a lot of what
is really going on.
Finally there was a team sport which involved speed, necessitated excellent coordination skills, nifty footwork, and you didn't need to be built like a brick wall. In fact some of the best international football (soccer) players in the world are under 1.80m, and if you recall Maradona, arguably the most charismatic and talented player since Pelè, is only 1.65 m tall.
While soccer grew in popularity also as a boy's sport, I believe the term, soccer mom became established. She could chaperone her children, sons and daughters, to games and training practice. She no longer needed to rely on her husband to teach their son to play a manly sport, she could kick a ball just as easily and score a goal, besides her husband probably knew next to nothing about soccer unless he had watched South American TV sports channels.
What is interesting is how the term, soccer mom, has, in such a short space of time evolved from being one of praise, to its present day derogatory meaning.
From Urban Dictionary, its most impartial definition (edited on my part)
Soccer mom
A middle-aged, upper middle-class woman (usually white) and lives in
the suburbs who devotes her life to her children. She carpools,
drives them to soccer and little league; volunteers at their school,
does snack days, and play dates. Most of them end up driving their
children away by not letting them express their selves and
immediately putting down anything that they find important. They are
usually Christian and this can be shown around their house, in most of
the cases I have found the children end up being atheists. They [soccer moms]drive
in their mini vans and suburbans with their fancy coffees and cell
phones.
That mom that is driving like a maniac to pick up her kid from
school and cart them to soccer practice is a soccer mom.
EDIT
For the downvoter. Why is the term called Soccer mom and NOT Basketball mom, Baseball mom or Hockey mom? Because soccer is a relatively new sport in the US, although its roots were formed as long ago as 1860, it saw a dramatic decline in popularity in the 1920s. Americans had to wait until the 60s before it began to regain support. American soccer girls teams in the US have existed since the mid 70s but only began flourishing in the 1990s especially when it was decided that the US would host the Fifa World cup in 1994. As I mentioned previously, initially, soccer was embraced by American girls as a competitive team sport, American men and boys still tended to view soccer as being a sports for wimps, extolling the virtues of (American) Football as the sport par excellence.
Wikipedia states:
Soccer in the United States
In 1967 there were 100,000 people playing soccer in the US; by 1984,
that number had grown to over 4 million. Girls high school soccer
experienced tremendous growth in playing numbers throughout the 1970s
and 1980s—from 10,000 in 1976, to 41,000 in 1980, to 122,000 in
1990.
The 1970s and 1980s saw increased popularity of the college game.
Women's college soccer received a significant boost in 1972 with the
passage of Title IX, which mandated equal funding for women's athletic
programs, leading to colleges forming NCAA sanctioned women's varsity
teams. [...]
The growth of the women's game during the 1990s helped increase overall interest in soccer in the United States. The number of women's college soccer teams increased from 318 in 1991 to 959 in 2009
Finally, the statistics in the Wikipedia article suggests that soccer's unstoppable rise in popularity is no longer confined among the middle-class youth, but has cut across gender, ethnic, income and class barriers.
The largest category of soccer in the United States in terms of participation is boys' and girls' youth soccer. Soccer is one of the most played sports by children in the United States. In 2012, soccer was the #4 most played team sport by high school boys, and soccer overtook softball to become the #3 most played team sport by high school girls. As of 2006, the U.S. was the #1 country in the world for participation in youth soccer, with 3.9 million American youths (2.3 million boys and 1.6 million girls) registered with U.S. Soccer. The number of high school soccer players more than doubled from 1990 to 2010, giving soccer the fastest growth rate among all major U.S. sports
EDIT 2
Until yesterday I was unaware that (American) football is not considered to be the national sport. Despite it being on the whole the most popular sport in the United States [and] "As of 2012, nearly 1.1 million high school athletes and 70,000 college athletes play the sport in the United States annually" That special honour is reserved for baseball.
But not all Americans would agree as the executive editor of Slate, Josh Levin, boldly claims, "If the United States had an official sport, what would it be? Baseball can call itself the national pastime until the sun burns out, but the correct answer is good old American football."
If you're into sports, I recommend clicking on the article, it has a fun map of the different official state sports in the USA including a few surprises too!