It is 1 in 20, and your friend's arithmetic is correct — he is just answering a different question than the one that happened.
There are indeed 400 equally likely (guess, roll) pairs, each with probability 1/400. But you did not ask for the probability of one specific pair like (17, 17). You asked for the probability of a match, and there are twenty pairs that match: (1,1), (2,2), … (20,20).
So the probability is 20 × 1/400 = 1/20.
The 1 in 400 figure is the answer to "what was the chance he would guess 17 and roll 17, specified in advance". That is a genuine and much rarer event. But nobody specified 17 in advance — the guess was whatever he said, and the interesting thing was that the roll agreed with it.