I actually bothered to look it up before I replied.
Here I found that a four-piece sectional weighs about 1050 lbs.
And since the original poster thought his couch had many (many) more atoms than it actually did, I wanted to use a large couch to illustrate.
The odds that a specific hand would be duplicated are exactly what would be predicted from that number.
The odds that two people in the history of the earth have had two random decks that are the same is a little smaller, itâs true. Pretty much completely immeasurably smaller, but a tiny bit smaller, yes.
The QI quote posted above is excellent:
If every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people has a trillion packs of cards and somehow they manage to make unique shuffles 1,000 times per second, and theyâd been doing that since the Big Bang, theyâd only just now be starting to repeat shuffles.
Thereâs a Charlie Stross novel, right there. Or yours, what with it being your idea and all 
Itâs also worth noting that, although thatâs all correct as a mathematical matter, as a practical matter:
Playing card decks may not start out shuffled, but in a predetermined order.
Even if immediately shuffled by hand, they are humans, with similar biases, and, although some randomness will certainly come from that, a good deal of non-randomness that people THINK is randomness will also come from it: cutting the deck in half and interleaving a few times, for example.
So itâs quite probable that, for varying levels of shuffled, a shuffled deck may have been duplicated several times.
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