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It's difficult for me to give a clear answer here because a) the question is pretty subjective and b) I'm not an expert in U.S. history. So I will focus more on Mexico in colonial times which I'm more familiar with, and give a broad overview of Spanish and native American influences that might have formed such a perception.

I'll quote quite a bit from this earlier post I wrote that seems connected, hope that's okay.

Spanish Influences

Spanish colonisation of what is today Mexico began with Cortés conquest campaign starting in 1519. These led to the fall of the Aztec Triple Alliance with the help of larger numbers of native allies; and to the beginning of the Viceroyalty of New Spain covering large parts of central America. Spanish influence in colonial Mexico meant among others the implementation of European burocracy and conversions to Catholicism. Here we are several decades before the larger-scale European colonization programs in Northern America of the late 16th century.

What is more, such Spanish colonisation often built on centuries-old medieval Iberian traditions (possibly reinforcing a perception of "historical depth"). Regarding differences between the earlier Iberian colonial empires (Spanish and Portuguese) and the later empires (incl. the British), Jürgen Osterhammel has highlighted the role of settler colonies: Briefly put, in the Spanish empire we see a comparatively high degree of "mixing" between different population groups, including European, native, African and Asian. This was reglemented through a hierarchical society, which I explain some more in the post linked above. Osterhammel contrasts this earlier form of colonialism with settler colonies that "functions through the replacement of indigenous populations with an invasive settler society that, over time, develops a distinctive identity and sovereignty". Examples of this type include Canada, the United States, but also Australia and South Africa.

To go into some more detail on how this casta system worked in Spanish America, I'll quote from the answer linked above [edit: although the casta system was in evidence not only in Mexico but also in other parts of Spanish America, I'll look at it here as it seems relevant to the question]:

This system was brought by the Spaniards to their overseas possessions. It had roots in medieval precedents, the "purity of blood" degrees used in Christian Iberia to distinguish between Christians and (converted) Muslims and Jews. Simply put, in the Americas the casta system was supposed to hierarchically distinguish between different social groups. María Martínez has argued that "in Spanish America, the notion of purity gradually came to be equated with Spanish ancestry, with “Spanishness”, an idea that had little significance with the metropolitan context. The language of blood and lineage also underwent modifications. Nonetheless, at the end of the colonial period, the concept of limpieza de sangre was still partly defined in religious terms”.

Casta groups included the Spanish at the top of the pyramid, who held the highest posts including that of Viceroy; and "indios" as mentioned, 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. The bizarre nature and the crazy amount of subdivisions in this "ideal type" system between groups becomes (hopefully) clearer in this casta painting. Such painting became current esp. in the 18th c., and were sometimes even commissioned by kings and viceroys, as in colonial Peru.

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.

So regarding your question, I would say that we have Iberian traditions playing a strong role in colonial Mexico; but also a mixing between population that is very different from the colonial U.S., and which has meant a stronger native influence on modern-day Mexican society.

Edit: A note for context