Why is the total number of atoms in the observable universe often considered as an upper bound for computational feasibility?
Let us assume an explicit range of the number of atoms in the universe to make sure that we are talking about the same thing.  It is estimated that there are between $10^{78}$ to $10^{82}$ atoms in the known, observable universe. In other words, it is a number between ten quadrillion vigintillion and one-hundred thousand quadrillion vigintillion atoms.

Is the number of atoms in the universe large enough?

The common unit of computing performance is one floating point operation per second (FLOPS, flops or flop/s).


How much is the total computing capacity worldwide today? According to a popular estimate, "computing capacity worldwide was probably around 2 x $10^{20}$ – 1.5 x $10^{21}$ FLOPS, at around the end of 2015". 
Extrapolating that by a generous 50% yearly growth rate, the total computing capacity worldwide today, around the end of 2018 is less than 6 x $10^{21}$ FLOPS.
Suppose you can run the total computing capacity worldwide today continuously for 100 billion years (which is much longer the life span of our universe as commonly estimated by physicists), you would have done a whopping 6 x $10^{21}$ (seconds) x 3.1536 x $10^{19}$ (Flop/s) $\approx$ 2 x $10^{41}$ FLOPs. 
Suppose everyone of about 7.6 billion people on the earth is enjoying the same computing journey and there are another 100 billion planets like earth, we would have done a total of 2 x $10^{41}$ x 7.6 x $10^9$ x $10^{14}$ $\approx$ 1.5 x $10^{65}$ FLOPs. 
However, that number is still much less than a billionth of the number of atoms in the universe! If each FLOP would count as 1, we need much more computing device and time to count the number of atoms in the universe!


Even given the impossibly optimistic yearly doubling of computing capacity worldwide, we would still need tens of years before our total computing ability to be anywhere near the number of the atoms in the universe.

Why choosing the number of atoms in the universe?

As Rick Decker indicated in his comment, that number is about the easiest way to say a big but fathomable number just about right to convey the message of practical un-computability. The vivid image of an immensely large universe filled with one of the smallest particles brings immediate understanding of an otherwise unfathomably big number to all walks of people. 

Another nice huge number, Google, $10^{100}$, is sort of void of concrete meaning.

Further Topics

Which project is the largest computing project that has been completed or is going on?

Does FLOP capture all kinds of computing power? For example, quantum computing?

Why is the number of atoms is a really very small number compared to the number of combinations that we have to deal with routinely in many practical computations?