While I agree somewhat with what you're saying, I'd like to ask something about the playing card example.
In a deck of cards, if you consider any ordering without any meaning added, then it is literally just 1/52!.
however, what if you started ascribing meaning to the different cards?
Would it be special then if your shuffeling resulted in all face-cards being grouped together? What about if all like-numbers were grouped together (i.e. all 1's, all 2's, all 3's) and so-on?
What about the case when the cards get shuffled and end up in ascending/descending order? i.e. {1H, 2H ... KH, 1S, 2S, ...}. There are only 4 such permuations possible, making it an infinitesimal chance that you arrive at that specific ordering.
Couldn't that be viewed as special? I guess in this case, I'm saying that the tuple of {Meaning, Uniqueness} is indeed special, not just uniqueness alone.