The most comparable game to chess that has been solved is checkers, where it has been shown to be a draw given perfect play by the second player. The first move gives a player a very slight edge initially, but does that convert to a winning advantage?
Tablebases give us some insight into the debate. The vast majority of positions which are materially balanced result in draws. Positions that are somewhat dynamically balanced (say knight vs bishop) still result in draws a high percentage of the time, although the more powerful the remaining pieces are, the more likely first to move is the winner. (take for instance, KQRKQR endgames, the first to move wins 67% of the time).
Another important factor seen in tablebases is that most wins have a relatively short distance to conversion (or moves to force a win). There are extreme cases, for instance, the record had jumped from 292 moves in 1989 to 330, and then to 545 in 2006. The striking quality to me is that the gap between move lengths in these records jumps so much all at once, which suggests that it gets harder to force a win the more moves away from the end you are, because most positions that far away are draws. It would be a winning lotto ticket if starting from the opening position, it happened to be one of those extremely long won positions. To me, this is strong circumstantial evidence that chess will be another game shown to be a draw.
Unfortunately, it is currently impossible to generate a full 32 piece tablebase for chess, as there are more possible chess positions than atoms in the universe. Barring some breakthrough in quantum computing allowing all possible positions to be evaluated simultaneously, I doubt chess will ever be fully solved by man.