Is social class "natural" or constructed?
In an everyday scenario, we all know that money does not buy class. It can purchase a good enough costume to make it look like someone has class, but behavior is often the key of whether one will be accepted or not in certain society.
In a scenario such as Daisy Miller's, her world is quite different than ours. As a nouveau-riche, Daisy has been able to basically "purchase" her ticket to join the high society. However, this high society we are talking about is quite different from what Daisy is used to knowing. For once, these high class individuals such as Winterbourne and his peers are closely related to the aristocratic classes, or have strong ties with their European ancestors. This is especially true of the early 20th century New York dutch-tied society as shown, for example, in Edith Wharton's Age of Innocence.
Due to these strong ties to Europe, the American nouveau-riches had the money, but not the connections to establish a strong peerage with powerful families. Hence, what many of them did was to become patrons to poor aristocrats that would benefit them with a European connection.
All this being said, Daisy still lacks all the elements she needs to succeed as a bona fide member of the upper crust of society: She lacks the needs that they have to preserve an image of superiority and power. Hence, while they display behaviors that date back to their proud ancestry, she displays the behaviors of, well, a nouveau riche. This is both shocking and annoying to most upper class people. This is also the reason why it was easier for Winterbourne to dismiss Daisy than what we as readers would have expected.