There are three cases in which it is easily possible to distinguish strong AI play from strong human play:
- The AI is playing at super human skill level
This seems obvious, but I want to mention it for the sake of completeness. The current skill ceiling of top level chess is well known and an opponent playing way above this skill ceiling must either be an AI or a chess guru who hid in a cave for the last centuries. Applying Occam's razor I would go with the AI. So to mimic human play the AI must make sure to stay at a human ELO level.
- The AI plays an obvious weak move in a losing or strange situation
When the AI is in a bad situation and still tries to win the game without any reasonable move available, it might be tempted to play an obviously bad move, because the decision engine assigns it the highest probability of success. Such moves are considered rude at high level play and a human player would not play them but resign. The most famous example of this is move 101 in the forth game between Lee Sedol and AlphaGo. AlphaGo just lost a big group but still played T9 to extend it even further. This is a move a rookie player stops making after his first 10 games, because it is that obvious that it does not work. A professional go player would never do this under normal circumstances. AlphaGo was out of options and played the rude move instead of resigning. This doesn't mean an AI cannot be programmed to avoid such behavior, but dire or simply very strange situations might induce moves like this, giving the nature of the AI away.
- Other factors than the actual moves are considered
Depending on the test scenario, the AI might give away it's nature through other means than the actual moves. Things like moving too fast or playing without breaks for hours. But I assume the question was just focused on the actual play and not the surrounding factors.
Those are the obvious examples where an AI might give away his true nature. I highly doubt that a strong AI, playing at strong human level, can really be identified because of his "inhuman" style. I think it is more of a psychological pitfall to consider innovative moves from an AI "inhuman" and the same move from a pro player as "brilliant", if you know who is who. I am an amateur 3dan Go player - far from the top but with several years of experience under the belt - and at least I wasn't able to see anything "inhuman" in AlphaGos play so far. In go it is well known that there are different styles of play, and after reviewing enough pro games one gets a sense of which player might have been educated in Korea, China or Japan, because certain patterns of strategies are more or less common in those countries. The lack of such an individual nuance might be an indication, that the player is actual a strong AI, but I am confident, that even the strongest pro wouldn't have a high success rate guessing who is human and who is the AI.
In chess there are significantly less options at every move, many fixed patterns especially in the opening and therefore less room for individual style. Therefore I consider the same challenge even more difficult than in go. So if an AI has reason to "play like a human", this should be possible with the current level of AI technology.