Chris Anderson is right: we should stop using the word "economy" as rhetorical smoke when we describe non-monetary markets and start measuring them like the economy they really are.
I'm not an economist, and many of the words used in the above mentioned post and the comments that follow it have a much more "common" meaning to me.
However, I feel that we are talking about the production and distribution of goods, right? There are many economic models and capitalism is just one of them (I get the impression that most Americans refer to economy and economics as a synonym to capitalism and free market...). Even capitalism has changed much since Adam Smith, mainly thanks to the industrial revolution. Land, labor and capital have changed a lot!
Gutenberg made written words part of the economy, by turning any form of written word into a good that could be traded and distributed. This is what the Internet is doing now: it is turning attention into something that can be traded and distributed.
Adwords is a way of trading accumulated attention (called "reputation" or "fame"), for money. It could be said that HTML links are a way of distributing attention. 
It's not something new -after all, books existed long before printing press, right? Celebrities have been doing this for decades. Brands, in general, too. But it took a lot of investment, a costly machine called mainstream media, to accumulate, distribute and trade attention, much like it took monasteries and monks and years of work to produce and distribute the Bible before Gutenberg.
During the past centuries innovation, patents, knowledge, leisure, art have become part of the economy. They have become goods, wealth, capital, labor, investment, (individual, corporate or national) they have been traded, distributed, produced. It took mechanisms like the printing press or copyright laws, the patent system or radio to get there. Now it's attention's turn to become an important part of the economy and the Internet is the mechanism.
I was reading  that, in 'Economic Possibilities of Our Grandchildren', Keynes states that "in the future" the problem of society will not be how to find leisure, but how to cope with unprecedented quantities of it. Like him or not, isn't this the 'Attention Economy'?
--  Intangible? Yes. But there is an interesting "economic" twist here. One may say that no matter how much attention (usually measured in pagerank units in the Google universe) I give, I don't become less famous. However, I have less attention to give! This is obvious in everyday life, but even for Google, the value of an outgoing link decreases as the number of outgoing links from a specific page increases.  The Worldly Philosophers: The Lives, Times And Ideas Of The Great Economic Thinkers [7th Edition]