Looking through history, we see terms that we now find anachronistic or even racist and bigoted that years ago were not only perfectly acceptable and common, but even used as a self-description by those minorities.
Over time, those minorities; racial, national, ethnic, religious, and even cultural, would take on those terms and use them internally as self-descriptors, but abandoned them later.
Essentially, it's the evolution of "Irishman", to "Irish-American." Also similar terminologies for blacks, different Christian sects, homosexuals, women, etc.
PLEASE NOTE I flaired this non-meta post as being posted by myself as a Moderator. Normally I do not do this, but due to the potentially loaded nature of the question and how it can (and likely will) invite poor comments, political rants, and even racism, I wanted you to know that this post will be monitored. As I said, normally I would never do this, but this is being done as a heads up.
In many cases, these were labels created and applied by the majority upon the minority, who had no say in the matter and were not asked. Or at least this is how those minorities perceived it.
In the case of Asian-Americans and the term “Oriental,” this happened after the Civil Rights movement, since large waves of Asian immigrants only came after the change in immigration laws in 1965. Writing in 1989, Ronald Takaki, in Strangers from a Different Shore, writes how younger Asian-Americans were thinking about this at the time:
"Today, young Asian Americans want to listen to these stories--to shatter images of themselves and their ancestors as "strangers" and to understand who they are as Asian Americans. [...] They want to know what is their history and "what is the movies." They want to trace the origins of terms applied to them. "Why are we called 'Oriental'?" they question, resenting the appellation that has identified Asians as exotic, mysterious, strange, and foreign. [...] Who decided what names would be given to the different regions and peoples of the world?"
In the case of Latinos, during the Civil Rights movement, Chicanos in the South-West began objecting to the term "Hispanic," which they also felt was being imposed upon them without their say in the matter. They especially disliked the fact that it identified them as having to do with Spain, while they themselves showed pride in their indigenous roots and their mestizo nature. Today, however, this all depends upon region and age. In most of the country, Latinos don’t really care if they are called “Latino” or “Hispanic.” Many actually prefer “Hispanic.” And lots of others prefer to be called from the country of their ancestors, e.g. “Mexican-American.”
Other changes such as “Frenchman” to “French,” probably just has to do with language change, with the former just sounding old-fashioned.
What I believe you're trying to ask is why the "other" or "outsider" in the U.S. was identified as such, and how/why this changed? For the purposes of this answer, I define anyone who is not white or of Anglo descent as the "other".
In the nation's history, black Americans have occupied the position of "other" for almost its entirety. The defining characteristics of African Americans popular in antebellum America changed after the Civil War when they gained agency through economic and educational growth. In the southern states, new laws restricted this development for AA's, allowing many negative stereotypes to persist.
Source: 1. Richard Valelly, The Two Reconstructions : The Struggle for Black Enfranchisement (Chicago: University of Chicago Press, 2004).
The evolution of the Irishman also developed during this period, as many were taken straight of the boat at Ellis Island and placed directly into the Union Army. Back home, the Irish were fighting for independence from England and many came to the U.S. to gain military experience that they could take home to Ireland. As the U.S. experienced the second industrial revolution, and urban populations exploded, the Irish earned greater acceptance in relation to AA's and Eastern European immigrants. Irish immigrants told friends they learned 3 things about America. First, the streets weren't paved with gold. Second, the streets weren't paved, period. Third, the Irish were expected to pave the streets.
Source: 1. Mitchell Snay, Fenians, Freedmen, and Southern Whites : Race and Nationality in the Era of Reconstruction (Baton Rouge: Louisiana State University Press, 2007).
As for the additional races and their acceptance, the best way to gain validity as a U.S. citizen was to buy a house. The reasons for this are many, including: it meant the individual was invested in the nation; owning a home required steady employment; homeowners were less likely to be radicals.
Source: 1. David Roediger, Working toward Whiteness : How America’s Immigrants Became White (New York: BasicBooks, 2005).
Hoped this answered some of your question.