Strictly speaking, we don't allow example-seeking questions on AskHistorians because they tend to invite disjointed and confused answers that don't live up to our normal standards. In this case though, there's a very interesting theoretical undercurrent to your question that I think means we can give you a quite specific answer.

What you're essentially describing here is a phenomenon that some historians call 'settler colonialism'. Broadly speaking, in academic discourse on European imperialism you will generally find that a distinction is made (either implicitly or explicitly) between two kinds of colonial project: the colony of occupation and the colony of settlement.

Colonies of occupation are those most of us probably think about first when we try to think of examples of European imperialism in the 19th century. They feature a process of colonialism whereby a small group of people from the colonising power (which historians call the metropole) occupies and exploits colonised territory at the behest and on behalf of that power. Although that group may come to develop a sense of somewhat unique identity based upon their own experiences in the colony, the vast majority continue to identify with the colonising power above all else; the colony is an extension of the metropole. The British imperial project in India is a good example of occupational colonialism at work, where white colonisers retained their distinct British identity and connections to their homeland, their fundamental interest lying in exploiting the people and resources of the Indian subcontinent in the interests of metropolitan society and government. The exploitation of Human and natural resources is the primary focus of occupational colonialism; the presence of colonising communities is about providing the infrastructure necessary to facilitate that exploitation.

Settler colonialism is a different beast. It is still marked by the exploitation of people and resources - these things are intrinsic to all colonial projects. What sets settler colonies apart is that the colonists themselves engage in the project of actively building a new society for themselves in the territory they colonise; they do not exist solely as an extension of the metropole for the purpose of serving its economic and political interests. Settler colonists sought to build a new future for themselves and their families in the territories they occupied; to use the resources and people they exploited for the benefit of their own developing societies, not just in the interests of the imperial power from which they originated.

One of the defining features of settler colonialism is that the settler seeks to displace - both physically and symbolically - indigenous populations. For the settler colonist, arguably the most important thing in the process of colonisation is the authentic and total control of productive land. Access to vast quantities of productive land presents unique opportunities for the acquisition of wealth, the building of new communities and liberating opportunities for advancement or growth that are not available in the more rigidly organised societies of the metropole or an occupational colony. The control of that land requires the forced removal of indigenous people from it, as well as the cultural legitimisation of that removal to justify it. Whilst occupational colonists broadly seek to exploit native populations for their labour, settler colonies seek their total displacement from the physical and social spaces they have decided ought to belong to them, and devise a myriad of ways to achieve that displacement (up to and including genocide). In the histories they write, this process of displacement becomes part of the process by which their entitlement to land is earned - narratives of arrival and surviving hardship, including difficult encounters with native populations, are often a key component of settler-written histories and mythology.

This obsession with land and expansion helps to provide settler colonists with a shared, unique identity. In the experience of being settlers forging a new future in what is to them a new world, settlers find a commonality that - to some extent - enables them to forge an identity that bridges divisions of class, gender and religion carried over from the metropole. The land itself becomes the thing around which their shared identity revolves; their exploitation of it is mythologised into a kind of national origin story for their new society, one which all settlers can ultimately identify with on some level. This national origin story in turn becomes a moral and political self-justification for the continued displacement and subjugation of indigenous populations. At the same time, there is a tendency for these settlers to push away from the metropole they leave; each successive generation identifies less and less with their country of origin, usually coming to see its government or society as somehow flawed and alien, and more with the land of their birth. Metropolitan power tends to become seen as a necessary evil rather than a desirable good - a mechanism by which the security of the colony from the threat of indigenous resistance is preserved, and a power that could support the colony in its ambitions of further expansion and exploitation.

What settler colonists essentially do is co-opt the language and hallmarks of indigeneity for themselves, and deny the legitimacy of actual native people's claim to that identity. As they come to associate themselves on a deep and almost spiritual level with the land they claim, they write the actual indigenous population out of the history of that land, recasting the settler as native and the native as outsiders. You cannot build a society and nation on the idea that you are an occupying force; to establish an authentic community of their own, settlers had to write their own national origin story. Often, part of this process involves some re-purposing of indigenous practice and tradition, so that the societies that create are fundamentally European but at the same time somehow different to metropolitan society. Settler colonialism is about building inherently European societies outside of Europe - but those societies must also be somewhat distinct culturally and socially from their mother societies to justify their own authenticity and claimed right to indigeneity.

There is some scholarly debate as to whether or not settler colonies ever become truly and authentically 'postcolonial' - that is to say, whether the period of colonialism has actually ended in these colonies, to the point where we can talk about the legacy of colonialism. Some scholars have made very powerful arguments that settler colonies today remain still very much colonial in nature, in the sense that they continue to exhibit what some call 'internal colonialism' - the continued exploitation and systematic marginalisation of native and minority populations to advance the interests and claim to sovereignty of the descendants of white settlers. It can certainly be argued that well into the 20th century and beyond these processes of internal colonialism, whether they're seen as post-colonial in nature or not, essentially kept fulfilling that role of reinforcing the identity and settler mythology of European-descended groups.

The 'classic' examples of settler colonialism are Australia and New Zealand, though Algeria is also commonly mentioned, as you mention in your post, although I'm not particularly qualified to talk at length about any of them, I'm afraid.

If you're familiar with South African history, it might be apparent by now though that these models of understanding colonialism aren't perfect - South Africa is a very good example of why they aren't perfect. The logical conclusion of settler colonialism is the overwhelming of indigenous populations by settler ones demographically and spatially; that has not happened in South Africa. Likewise, whilst the widespread displacement of indigenous people was absolutely a feature of South African colonialism, the peculiar way in which South Africa developed also has distinctly occupational characteristics. The homeland system of the apartheid era and the exploitation of modern-day Namibia, for example, could be conceptualised as an attempt by white South Africa to reinvent itself as an imperial power in its own right with the homelands as colonies for exploitation by a new metropole. The argument can be made that whether South Africa is an occupational or settler colony, and at what point (if any) it has become a post-colonial state, depends on whose perspective you look at. There are similar problems with conceptualising the United States.

For the purposes of your research interests though, both South Africa and the United States can be understood as settler colonies. White European identity in these entities was fundamentally forged through the same process that it was in Australia or New Zealand, focused on land, displacement of native peoples and the co-option of the claim to indigenous identity. Certainly in both countries the rhetoric of popular history and national mythology has traditionally celebrated the ideals of settler colonialism from the perspective of the settler.

I hope that this makes sense and has been helpful to you. As I said at the beginning, we don't generally allow example-seeking questions - but hopefully this answer has gone some way to helping you understand the process of white identity formation in places like South Africa, which seems to be what you were essentially interested in judging by your interest in integration and assimilation.