Any professional go player can remember their own games as matter of course. They also replay hundreds of pro games for study, thereby memorizing them. I'd be surprised if it were any different for chess players.
In fact, even amateur low-dan -- advanced but far from professional level -- players can replay games they have played in the same day, and can memorize a few games with some practice.
Go games often last more than 200 moves but players memorize sequences, not isolated moves.
Pros can routinely read 50+ moves ahead but, as tsipiteri mentioned, the hardest part is evaluating the positions.
While there is no single "world" pro championship, a couple of Korean teenagers have won prestigious international titles. A 19-year-old who's "merely" a contender wouldn't be called a prodigy in the go world. I guess Asian go players just train more and are more competitive than chess players. Unlike Carlsen, they don't usually care about a life besides their game.