This question comes up from time to time and has a section in the faq.

The most interesting answers for your question from there might be:

• A great overview by u/Yawarpoma

• This one going into Spanish American organisation.

• For a more specific example, I've written here about "mixing" in Mexico and about the casta System in Spanish America.

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I'll highlight a few important points below focusing more on Spanish America which I'm more familiar with (adapting this from an earlier answer):

Importantly, looking at demographics, a larger part of the pre-hispanic native population was concentrated in Central and South America in comparison to North America. This was influenced but not "reversed" by the demographic catastrophes that struck native people in Mexico and other places post-conquest (this is discussed in more detail in the 1st two answer I linked to).

Differences in the time-frame play another important role here, as the ealier colonial powers were organized quite differently than the later colonial empires. The settler colonies - e.g. of the British in N.America and many other regions - focused on extracting resources and had a stricter seperation between European settlers and native populations. Whereas e.g. the earlier Spanish and Portuguese empires had a stronger focus on using indigenous people for (often forced) work from very early on intermarriages between native, European and African populations. What is more, even already in the 16/17th centuries the French and British studies what they perceived as "failures" of the Spanish to control their overseas possessions, and modified their own legitimations accodringly.

So when a certain European power started colonizing is one main factor here. This also shows for example with the French empire, where in the early modern period "métissage" or mixed marriages where quite common compared to , say British North America. But where e.g. in a later colony like Algeria in the 19th century divisions between population groups were more rigidly enforced.

With social organization more specifically, we can also a more fluid approach at least in colonial Spanish America compared to the later stricter approaches in British America (e.g. with the One Drop Rule). I'm recycling my post I mentioned above for some insight into the socalled casta system:

Casta groups included the Spanish at the top of the pyramid, who held the highest posts including that of Viceroy; and "indios" ..., who were under the Crown's protection if they converted, and if noble could continue to hold a certain influence. Others were the "creoles" or Europeans born overseas; "mestizos" of mixed European-native origin; people of African origin; and Asians (often from the Philippines). Then we have a large variety of casta due to mixing between the groups (hopefully answering your last question). The important thing to understand here is that this was an *ideal * system which did not actually work completely as envisaged by the Spanish. ...

So we have at least in earlier colonial times (ca pre 17th c.) still quite of lot of flexibility between these groups. We know of noble Indians and mestizos who described themselves and were perceived as "Spanish". On the other hand, noble Spaniards starting with conquistadors tried to marry into the native nobility in order to gain legitimacy and privileges. R.D. Cope has spoken of "the limits of racial domination" in this context, meaning that even commoners of lower castas had a certain agency and could swith between groups. Nonetheless, he argues that there were long-term effects to the casta system and its exertion of elite Spanish control; and that it would influence people's perceptions of these groups.

While I can't go into more detail for the British/French on this, it seems to me that (esp. in modern times) there would have been less opportunities for a) intermixing between the various groups, and b) changing between the groups there.

Last but not least, I'd add that the different legitimations tied to judicial processes used by the Spanish and the French/British/etc. played another role. For the Spanish, the native people's conversion was central to their rights to colonise. This influenced their more long-term approach of conversion and cohabitation, as the native people when Christian where supposed to be under the Spanish Crown's protection. In contrast other powers like the British legitimised their colonisation more strongly with their rights to supposedly "uninhabited" lands (although this approach was also formulated earlier by the Spanish scholar Vitoria). This made it in practice somewhat easier to argue that a certain area was uninhabited and start conquering other areas from there, displacing the original inhabitants rather than attempting to convert them - even though there were many exceptions to these broad patterns.