First let's answer this question.
What counts as a non-human "user" of the internet?
Any electronic device connected to the internet. There are numerous types of devices that can be connected to the internet. Be it a laptop, printer, iPod touch, server, cellphone, tablet, television, google glass, desktop computer, etc....
A non-human user of the internet is simply a device connected to the internet.
Where do they get that claim from?
Their claim seems to be extracted from an article from CISCO, How the Next Evolution of the Internet Is Changing Everything.
In CICSO's article, a description image has been posted:

And it has been explained:
Explosive growth of smartphones and tablet PCs brought the number of devices connected
to the Internet to 12.5 billion in 2010, while the world’s human population increased to 6.8 billion, making the number of connected devices per person more than 1 (1.84 to be exact) for the first time in history.
...
Looking to the future, Cisco IBSG predicts there will be 25 billion devices connected to the Internet by 2015 and 50 billion by 2020. It is important to note that these estimates do not take into account rapid advances in Internet or device technology; the numbers presented are based on what is known to be true today.
...
Additionally, the number of connected devices per person may seem low. This is because the calculation is based on the entire world population, much of which is not yet connected to the Internet. By reducing the population sample to people actually connected to the Internet, the number of connected devices per person rises dramatically. For example, we know that approximately 2 billion people use the Internet today. Using this figure, the number of connected devices per person jumps to 6.25 in 2010, instead of 1.84.
....
Is this based in reality?
You need to understand that it is an estimate, it might not be fully accurate. However, it is very reasonable as they explained here:
In January 2009, a team of researchers in China studied Internet routing data in six-month intervals, from December 2001 to December 2006. Similar to the properties of Moore’s Law, their findings showed that the Internet doubles in size every 5.32 years. Using this figure in combination with the number of devices connected to the Internet in 2003 (500 million, as determined by Forrester Research), and the world population according to the U.S. Census Bureau, Cisco IBSG estimated the number of connected devices per person.
...with an interesting reference:
While no one can predict the exact number of devices connected to the
Internet at any given time, the methodology of applying a constant
(Internet doubling in size every 5.32 years) to a generally
agreed-upon number of connected devices at a point in time (500
million in 2003) provides an estimate that is appropriate for the
purposes of this paper. Sources: “Internet Growth Follows Moore's Law
Too,” Lisa Zyga, PhysOrg.com, January 14, 2009,
; George Colony, Forrester Research founder and chief executive officer, March 10, 2003,