The question asked didn't mention the World Bank or any other definition of poverty, at least not judging from the results presentation. The World Bank, incidentally, has multiple definitions of extreme poverty, although all of them are lower than reasonable industrialised-world thresholds.

When around half the world's population's weekly purchasing power is, according to official World Bank approved statistics, equivalent to the purchasing power an American obtains after a five hours work on federal minimum wage[1] I don't think that Americans are unjustified in not knowing the "correct" conclusion that numbers or proportions of people in extreme poverty have "nearly halved" in recent years. If you think the unspecified extreme poverty bracket ought to include the ~50% of the population still stuck at $5 per day, it would have been literally impossible for the proportion of people in that bracket to have already halved in recent years.

I'd have accepted that it might appear a bit ridiculous for them to think that extreme poverty had doubled - especially given that it is so pervasive - except... they could have been inferring it from a doubling of "extreme poverty" reported in some studies focusing on their own country[2]. I assume results of papers like this get more mainstream coverage in US media:
http://npc.umich.edu/publications/policy_briefs/brief28/poli...
So perhaps some Americans don't think the world extends beyond the boundaries of their country; perhaps as you suggest they guessed on a multiple choice question based on having seen lots of reports on epidemics and war in the news. As I said, I'm the last person to defend the wisdom of Americans or any other type of crowd. But frankly, it's no less ignorant to insist that everybody should know "extreme poverty has halved" when the basis for that "fact" is predicated on the assumption that people earning $1.30 a day no longer qualify as extremely poor, even if other factors like working conditions have got worse. They're certainly right not to take that stat at face value.

We know medicine and access to it is getting better and that's having effects in lives saved. We know some of the GDP per capita increases are trickling down, even though our methodology for estimating how that affects the base of the pyramid is necessarily imperfect. But I don't think we're anywhere near the level of confidence of sustained global reduction in absolute poverty - as opposed to amelioration of its most acute symptoms - to be in a position to start celebrating how few people are really poor, never mind criticising the public for not being aware of it.

Not when - I really can't stress this enough - most of the world's population still have to survive for a week on the notionally equivalent purchasing power of an American working less than a full day's shift at Walmart.[3] That proportion certainly hasn't "nearly halved" and it's really not stupid or ignorant to think that the label "extreme poverty" could still apply to all those people

[1]if you believe PPP calculations, which are inadequate for this sort of thing since the weighting basket tends to include luxuries and so doesn't account for how cheap food and living has to be in most of the developing world. But again I don't think the inadequacy of measurement detracts from my point about poverty statistics being very much open for debate.
[2]using a different World Bank definition of extreme poverty.
[3]It's not that there's no validity to arguments that these people aren't extremely poor; I just don't see any reason why the argument that some of them are not should form the basis for widely accepted "facts". I've met people on that level of income in developing countries I wouldn't consider extremely poor of course. Then again, I've also met people literally killing themselves to earn $10 a day who I wouldn't consider anything other than extremely poor, even after accounting for it being considered a good income locally. Adjusted income-based measures of poverty don't really have anything going for them other than relative convenience.