I can't speak for the Spaniards but I can offer some insights with regards to Portuguese colonisation. It's likely that this will also apply to the Spanish but I can't be sure.

Much of what I say comes from Sérgio de Holanda's Raízes do Brasil (Roots of Brazil), a fascinating study of the very beginnings of Brazilian civilisation with a lot of information about the Portuguese as well.

Firstly, de Holanda argues that the notion of racial purity was never really a big thing in Portugal. Not only had the country been invaded by very different groups throughout its history, but it had been under Moorish rule for centuries. Its proximity with North Africa also ensured that the Portuguese were 'accustomed' to racial differences and mixed marriages. It is not very logical for a civilisation to look down on another race if they themselves have mixed with other ethnicities for a while.

Secondly, Portuguese migration to Brazil was for a long time very 'masculine'. Not a lot of women emigrated alone, of course, which led Portuguese men to find partners amongst the natives, many of whom would already be of mixed Portuguese-Amerindian heritage. A point of mine that I would add here is that due to the Portuguese phenotype it is very easy for a mixed race Brazilian to pass as Portuguese, even in those times. The same, I believe, would not apply to mixed British or Irish people.

Thirdly, and this might be very specific to the Portuguese, de Holanda also makes a point that the native indians were seen as somehow superior to Africans, with one tribe leader even being awarded a noble title by the King of Portugal. Now, the reason for this does sound a bit...odd, though de Holanda was en established academic: he says that the Portuguese identified in the native Indians the European trace of just not doing much in the way of hard labour. The African population was, of course, forced to do the harder jobs. A point worth noting here is that the idea of lighter skin being somehow 'superior' in Brazil stems in great part to the fact that it indicates the individual did not have to engage in hard labour (which is out in the open under the scorching Brazilian sun).

Further, the Portuguese have depicted Brazilian Indians in a relatively decent light since their first contact. Pero Vaz de Caminha wrote to the King of Portugal to notify him of the discovery of Brazil, and he seemed to be particularly fond of the Amerindian women, whereas the men were described as docile and innocent. The Indians also seemed to accept Catholicism somewhat easily, which put them in the Portuguese's good books: in fact, one of the first acts by Cabral's company was to put a huge wooden cross in the new found land.

All of this contributed to the notion spoused by Gilberto Freyre, that the Portuguese were naturally the perfect fit to colonise Brazil due to their 'Mediterranean' roots and their disinterest for racial purity. This last view is, expectedly, very controversial but worth stating nonetheless.

On the other hand, it is also worth noting that virtually every migrant community that came to Brazil mixed with the natives. I say this just so as to dispel a possible myth that Southern Europeans are more prone to mixing with Amerindians. The Dutch, for instance, had a colony in Brazil for a couple of decades in the 18th century and you can find remnants of Dutch DNA in Northeastern Brazil to this day.

In conclusion, I think it was a mixture of a natural attraction towards the natives (who are closer to the European standards of beauty than, say, Indians or Black Africans), a genuine lack of women in a time where a lot of Portuguese men were trying to make a living in Brazil, and a bridgeable cultural (and physical, related to phenotype) gap that the native Indians seemed good at crossing. This is not to say that the Portuguese didn't mistreat the Indians, mind you, but in general they were looked at more favourably than their African counterparts.

So I can speak for the Spanish. First, there really isn't much of a 'national' preference for or against mixed race unions. That is English, Spanish, Portuguese etc. don't inherantly eschew mixing per say, more importantly I would argue are the demongraphics, political-economy, and ideologies of expansion that shaped the creation of colonies.

So for the Spanish... Spaniards involved in conquest did not really plan to stay in the Americas. Most conquistadors were men of middling status (i.e. not poor farmers, not aristocrats, not wealthy professionals). They had assets to pay for passage to the Americas and to equip themselves as conquistadors (there was no royal army of conquest). They participated as private entrepreneurs. They hoped to make it rich with spoils of war, return to Spain, and live of those spoils and move up the social ladder into the petty nobility or if really lucky the titled nobility. This of course did not happen for most conquistadors.

But this did shape the flow of Spaniards to the Americas. During the first century of conquest and settlement, only about 25% of total immigrants were women and less than half of that were unmarried young women. This meant that there was a very real problem of reproduction. There simply were not enough Spanish women for all the men living in the Americas.

This meant that Spaniards wishing to marry, or at least have female partners, needed to find those among indigenous women, free or enslaved African women, or women of mixed race, mulatas or mestizas. Of course, not all of these relationships were consensual. Many Spanish men kept indigenous or African concubines forcibly. But others, especially as prospects of a Spanish wife waned did form consensual marriages/unions with non-Spanish women.

These mixed unions born of the earliest unions/marriages led to the creation of a large population of individuals of mixed ancestry. The Spanish created a whole host of labels for these individuals (mesitzos, mulatos, zambahiagos, zambos, pardos, etc.) Overtime, these individuals formed unions with each other as well as with Spaniards, indigenous people and Africans which perpetuated a society composed of many groups of varied ancestries. For the most part in most places, individuals tended to be categorized as espanoles, indios, negros, mulatos, or mestizos. Terms like pardo, moreno, zambo, zambahiago were used in particular places at different times mostly to differentiate individuals of varying degrees of mixed African ancestry. On the Spanish-indigenous side, most people of mixed ancestry became mestizos or castizos if they had more Spanish than indigenous ancestry.

The Portuguese had similar problems, few female immigrants, but their colony was more shaped by the rise of a plantation economy. In general sugar producing regions of the Americas developed a tripartite racial order. With whites at the top, a mulato population composed of slaves of mixed ancestry, the children of white masters and female slaves, as well as free people of mixed ancestry freed by masters or the descendants of those freed. Then at the bottom the largest percentage of the society were African slaves. In places like Brazil and much of the sugar producing Caribbean most of these slaves were African born, many living only a few years before the high mortality of plantation slavery took its toll. A smaller subset of the slave population was composed of creoles, born in the colony. In many places creole slaves held more privileged positions, receiving training in trades or artisan production. Some served as overseers and others served in the 'big house' as domestics for the plantation owner and his family. This racial order was common in places like Haiti, Barbados, Jamaica, late colonial Cuba, etc.

The US developed differently for several reasons. 1) In general the English wanted territory to settle and make productive through agriculture. The Spanish wanted to conquer people to exploit their labor and extract wealth, rather than take land to work themselves. 2) Although male heavy immigration was common, it wasn't as pronounced as in Spanish and Portuguese colonies. The gender imbalance narrowed more quickly, and there were more poor white immigrants. 3) In the southern colonies, the 'one drop' rule quickly categorized anyone of mixed African ancestry as 'black' as a result, 'mulatto' 'octaroon' etc were not as common. They still were part of the lexicon but had no real legal value. The 'one drop' rule appeared in North America largely as a means to divide poor whites from enslaved blacks, in early Virginia there were moments when those two groups threatened the hegemony of the wealthier settlers. An increasingly white-black boundary served to prevent solidarity between two economically exploited groups. (The diversity of racial categories in Spanish America served a similar purpose, although through a different approach, as each category carried with it different legal and social rights and obligations.) 4) Although English-Native American unions happened, they were far less common than in Spanish America (where the majority of subjects were always Native American). The children of such unions could be referred to as 'mixed breeds' but they could also 'disappear' if fully integrated by their white families, or they lived at the edge of European society where their racial status was less pronounced.

For great works both new and old see:

Jordan, White over Black

Morner, Race Mixture in the History of Latin America

Morner, "Economic Factors and Stratification in Colonial Spanish America with Regard to Elites"

Morgan, American Slavery, American Freedom

Mattoso, To be a slave in Brazil, 1550-1888

Schwartz, Sugar plantations in the formation of Brazilian society : Bahia, 1550-1835

Klein, Slavery in the Americas: a comparative study of Virginia and Cuba

Kiple, Blacks in colonial Cuba, 1774-1899

Blackburn, The making of New World slavery: from the Baroque to the modern, 1492-1800

Sweet, "The Iberian Origins of American Racist Thought"

Sweet, Recreating Africa

Bergad, The Comparative Histories of Slavery in Brazil, Cuba, and the United States

Lockhart and Schwartz, Early Latin America

Martinez, Genealogical Fictions

Restall, Black Middle

Schwaller, Generos de Gente in Early Colonial Mexico

I think your premise might be wrong. There was plenty of "mixing" in British North America--lots of people who call themselves Indians today have some White blood--as well as constant complaints by the colonial authorities that people of European descent had gone native. And I don't think there's evidence of a unified policy on this stuff on the part of any colonizers.

At the risk of getting Jared Diamond-y, it's probably worth looking at the social and political structures of those local populations for at least some of these answers. In Mexico and in the Andes, in particular, there were central governments that already controlled large populations of peasants and slaves. The Spanish "only" had to conquer the armies of those governments to take over the entire state and put themselves at the top of that existing system. Whereas in North America, the Caribbean, or the Southern Cone, local government was both less hierarchical (no large pre-existing subjugated populations) and less unified (no single decisive conquest) and so it took hundreds of years of warfare for Europeans to become undisputed masters of the land. A natural consequence was that native populations mostly survived the conquests of Aztec and Inca empires, and mostly did not survive the slow, bloody, piecemeal conquest of the less-organized areas elsewhere on the continent.

Not trying to answer or insinuate an answer but I am wondering if a response can also address the dynamics of organization in the native cultures. For example how did the Aztec and Incan empires play a role in this acceptance versus other tribal areas that were perhaps not as well organized/oppressed. I recognize that the Aztecs in particular were also in a severe "down" period when first contact occurred.