From my experience (small to medium central European Opens), offering a handshake without words is a commonly accepted form of resignation.
The handshake is not part of any official rules. However, there is some reasoning behind it:
You shake hands after the game ended (just as you do before it starts). So you only start extending your hand once that end is "set in stone".
If you extend your hand without a concrete reason on the board for an end of the game (checkmate, stalemate, 3-fold repetition etc.), you silently imply the other possible reason: That you just resigned, which is possible unilaterally. The "silently" part is the reason why this is done this way: It's less disturbing for neighbouring boards than saying "I resign" or laying down your king (in many piece sets, you can't make sure that it doesn't start rolling around).
If you instead wanted to only offer a draw, the game would not have ended (yet), as your opponent still has to accept. Thus, extending your hand makes no sense yet. But conversely, if you (properly by words) offer a draw, now your opponent may offer a silent handshake! Because once they (silently) agree to your draw offer, the game is over, and they can indicate this by stretching out their hand.
Of course, this "implied" way of ending games, even if it is more polite to other ongoing games, has the drawback that lead to the question here: It requires that both players are on the same page of what is actually implied. Even more problematic than the question are for example cases where one side saw a 3-fold repetition and implied "draw", but the other didn't see the repetition and assumed "resignation". Probably not the case here, it sounded like neither mentioned anything else but resignation or free draw offer.
Edit: Because the question came up why the players would act this way, there are some thinkable scenarios:
- Black was dishonest. He deliberately acted ambiguously to trick White (in the hope that the arbiter would be convinced of his story that he only wanted to offer a draw). (correct result: 1-0)
- Black was very inexperienced. Sometimes, weaker and especially younger players have the habit of always offering a draw before they resign (sometimes no matter how lost their position is), in the vague hope that the opponent is nice or doesn't see how winning they are. The handshake genuinely was intended to be a draw offer that White misinterpreted. (correct result: 1-0 with a degree of uncertainty. In my opinion, White should not be punished for acting in good faith, while Black with certainty did not offer a draw the correct way and thus has the weaker case).
- It was actually White who was dishonest. We don't know the position. Maybe it wasn't actually all that winning and White exploited Black's naivete of not explicitly offering a draw by words. (correct result: 1/2-1/2)
All these cases are mostly applicable to amateur tournaments. Top (and even advanced) players know that they just look like fools if they offer draws in lost positions or try to cheat that way.
To know which of these cases applies, we need more information (for example, what was each player's rough evaluation of the final position, how reasonable can we expect both players treat handshakes as resignations/draw offers? etc.).