What do Americans call a 'double-barrelled surname'?

It seems "double surname" is more popular than "double-barrelled surname" in the US:

Is the attitude of society to such names changing in any way?

Well, those two countries have different starting points. The use of double- and other compound-surnames among Spanish speakers, and those from pre-Columbian tribes is very different to those among British, and they are more often found in the US than the UK, as are plenty of other cultures with different approaches to surnames. In turn, since there are lots of different approaches taken to surnames, there is less need for new immigrants to change their approach to surnames to match that of the country the immigrate to.

As you note, there is an increase in people who would themselves have been given simple single patronym as a surname to give their children a double matrinym-patronym surname, and this happened in the US earlier than in Britain and Ireland.

From the other direction, the US didn't really have a sense of a "real double-barrel name" among the aristocracy because they don't have an aristocracy, while the British had. A couple of decades ago, having a double-barrel surname in Britain that wasn't one of an old family might make someone seem like a pseud to some (aping the aristocracy but not being an aristocrat), but these days that attitude would be rare—nobody using such a name cares about the aristocratic history, and anyone who still cares realises that nobody else does.

I think it depends on what phenomenon you are referring to, but we call the sort of thing where the bride or groom or both stick their surnames together a hyphenated surname. The term is roughly as popular as double surname in American English.

If you mean the practice that's been going on for a long time in several European cultures, I think if we Americans do know a word for it we would call it a double surname. But I qualify that because many people are not familiar with the practice.

To answer your title question, we call them hyphenated names (if they are) or last names (if they are not). Hyphenated names are, in my experience, declining, being replaced with women either keeping their maiden surnames as middle names (e.g. Sarah Kennedy Gunn) or keeping their maiden names and forgoing altogether the husband's name (Sarah Kennedy). A percentage (not the majority) are giving their children their maiden names as their middle names. Sarah Kennedy Gunn likely writes out her entire name, but her child Michael will write his as Michael K. Gunn, and her daughter will likely identify herself as Hope Gunn Johnson when she marries.

If 'increasing numbers of mothers are not marrying the father of their children' how can double names be growing? Unmarried mothers have an incentive to either give their child the father's surname (for child support and legitimacy reasons) or to avoid doing so (when custody might be disputed and there is no interest in child support.) But these mothers rarely hyphenate their child's surname.

I think we're on a 20 or so year lag time following the ups and downs of feminism. As it's increasing again now in the US, there will likely be some new change in 20 years or so, probably women will no longer be taking their husband's surnames at all is my guess.