Not the way you describe it, no. If my last 5 plays are "RPSRP", and my next play is scissors. but looking at these previous five plays, both P and S would be the best plays against this range (2 wins out of 5). However the correct next response predicting the pattern is R (which does the worst against this range with only 1 win in 5).
Edit: Ok re-reading your initial post I think I misunderstood you. However it still doesn't explain how it could pick up the pattern quick enough - there are 3 possible responses each time, so until it's tried all 3 responses it won't know which is best given the fingerprint. However it will pick up the pattern by the 2nd cycle.
I think it's using a more sophisticated learning strategy than you suggest.
The "fingerprint" is used to search a "history" of games against other humans. Based on the move that most commonly follows that "fingerprint" the computer selects the winning move. I'm assuming the "history" comes from within the game itself which is probably played by lots of people like you who want to test it with one of the two possible cycles (RPS or RSP). I suspect that the reason it picks up on the cycle so quickly is because all the curious tinkerers have biased the data it uses to make moves.
Edit: Ok re-reading your initial post I think I misunderstood you. However it still doesn't explain how it could pick up the pattern quick enough - there are 3 possible responses each time, so until it's tried all 3 responses it won't know which is best given the fingerprint. However it will pick up the pattern by the 2nd cycle.
I think it's using a more sophisticated learning strategy than you suggest.