The United States had much more economic mobility in the 19th and early 20th century than most of Europe, owing mostly to an abundance of employment opportunities. Modern soundbytes like the USA being a nation of immigrants or having an entrepreneurial spirits were actually a symptom instead of a cause of the financial opportunity available in the United States.
After the Virginia colony failed to create a neo-aristocratic cash crop economy like that of the Caribbean islands, they transitioned to a new revenue scheme - the colony's administrators sold land to settlers either in exchange for money or for years of labor. This model later became widespread throughout many of the 13 colonies. The abundance of fertile land in North America spawned what economic historian David Carlton calls the "American agricultural ladder". Unlike in Europe, tenants in the United States could reasonably expect to save money to own their own farms by their late 20s and early 30s.
This meant the United States, even at the time of the American Revolution, already had a higher GDP per capita than any European country. Its people on average were several inches taller than the average European since they could afford better nutrition. The abundance of land also created a culture of ownership - at one point in the early 1800s, over 80% of white males in America owned their own businesses, and the majority of those businesses owned the buildings and land they operated out of. At at time in the Western world where property was the primary indication of wealth, this represented a very good economic success rate for the average person.
More importantly, ready abundance of land led to sever labor shortages in both cities and rural areas, which fuelled capital-intensive business models, innovation, and better returns per person. The United States was far from the only industrial giant by the early 20th century, but it was the only one by the 1930s to make widespread use of the assembly line and scientific management. Other nations with cheaper labor pools like Germany, the United Kingdom, and Japan made use of more labor intensive and quality-focused industrial methods, while the United States' factories were famed for their standardization and quantitative output. It was not just mechanical innovation that labor shortage combined with a ready availability of resources facilitated - commercial innovation occurred as well. The United States had some of the first truly integrated industries in the world, and a lot of modern management and business theory originated from American industrial groups during the age of tycoons. Industrial combination, consolidation, and monopolization were just another way to exploit the maximum number of resources with the minimum number of people.
Labor shortage also expressed itself in rural economics - the United States among all major industrial powers had the highest average acreage per farmer, leading farmers to more readily mechanize their lands and to plant fewer labor-intensive crops and more capital intensive ones.
These factors made the United States a very open labor market - it was much easier not to be "trapped" in a profession in the United States than in Europe. New immigrants could readily find work, and while they faced discrimination, labor was always needed. Culturally, this meant that traditional barriers (class, manners, education) to finding certain employment or succeeding in selling one's product were far less important in the United States than in Europe.
Perhaps most importantly, workers in the United States had a far higher level of disposable and investable income than in Europe. By the early 20th century, wages in the United States were at least twice those of most other industrialized nations, while prices for basic goods were only marginally higher. Differences like this on an individual level matter. After covering subsistence needs, the rest of the money of a family goes either towards savings, investment, or acquiring increasingly complicated products. The last three needs are vital to developing a complex consumer market, and increasing generational wealth. For example, someone making $100 who spends all of it on food, bills, and necessities accumulates nothing, while, assuming a 4% rate of return, someone making $100 who invests $10 per year will end up with $13,000 by the end of a 50-year career.
However, there are many caveats to this conclusion. The first and most obvious is race - opportunity was greatest for native-born whites, and the further you went from that profile, the lower upward mobility was. Second, wealth inequality in the United States was just as high, and in some cases even higher, than in most European countries. In spite of there being more opportunities and higher wages for the average person, much of the wealth in the United States until the 1930s was concentrated at the very top. This was due to the huge economies of scale advantages in a very capital-intensive market like the United States. As John Jacob Astor said, "the first $100,000 was difficult... after that, it was very easy". At one point, John D. Rockefeller alone owned 2% of the US economy. So, while spectacular rags to riches stories were possible, small riches to big riches stories were more common.
Finally, we need to be careful about how we define class mobility. The American Dream was in some ways a reality - an industrious commoner had a better chance in the United States than in Europe of eventually owning his own property and leaving a sizable inheritance to his children, and immigrants to the United States could expect to make far more than they made in their home countries. This is a definition of economic mobility in absolute terms, in which case the United States definitely earned the title of the land of opportunity.
However, some economists, especially in recent years, define economic mobility in relative terms, as in "how likely am I to become richer than my neighbor whose father was a CEO?". This has led some economics, most notably Seymour M. Lipset and Reinhard Bendix, to conclude that economic mobility is the same in all industrialized countries and only affected by income inequality. If you read into this argument, you soon realize it is very tautological. If we define economic and class mobility not as "how likely am I to be richer than I was", and more as "how likely am I to be richer than other people", then obviously mobility is the same in all countries with similar amounts of inequality. This has led to somewhat ridiculous conclusions like studies reporting low economic mobility in early 21st century China. In this case, the US in the late 19th century would not have fared any better than much of Europe.
TLDR: Because of a ready availability of cheap land and resources, there was a chronic labor shortage in the US compared to Europe, which led to higher wages and levels of industrial innovation. This led to huge wage differences between the US and Europe which facilitated familial savings and investment, and the buildup of more generational wealth. Cultural explanations for mobility in the US, such as an "entrepreneurial culture" or "no nobility" were mostly by-products of the ready availability of work and entrepreneurial opportunities, and higher levels of investment, instead of the causes of mobility. However, while mobility was higher in the US, it was still a very unequal society until the 1930s. These 2 analyses put together mean that the average resident in the US could expect more of an increase in wealth at the end of his life than the beginning compared to the average European, but they had largely equal chances of ending up ahead of someone who was born into a rich family.
Sources:
Carlton, David. Mill and Town in South Carolina, 1880--1920
Kaeble, Hartmut. Social mobility in America and Europe: a comparison of nineteenth-century cities. <This one I use purely to reference the work of Lipset and Bendix - this study actually concludes that economic mobility was not that much higher due to its use of the relative and not absolute definition of mobility. It's an interesting read from the other perspective, however.>
Engerman, Stanley. Review: Up or out: Social and Geographic Mobility in the United States.
Grusky, D.B. American social mobility in the 19th and 20th centuries.
Seavoy, Ronald. An Economic History of the United States: From 1607 to the Present.
Follow up question (please don't delete) how influential were nobility titles on economic upward mobility in the Europe of that same time?