About those claims- this is from Russel and Norvig, 3d ed. (from 2003, so a way back):
Go is a deterministic game, but the large branching factor makes it challeging.
The key issues and early literature in computer Go are summarized by Boozy and
Cazenave (2001) and Muller (2002). Up to 1997 there were no competent Go
programs. Now the best programs play most of their moves at the master level;
the only problem is that over the course of a game they usually make at least
one serious blunder that allows a strong opponent to win. Whereas alpha—beta
search reigns in most games, many recent Go programs have adopted Monte Carlo
methods based on the UCT (upper confidence bounds on trees) scheme (Kocsis and
Szepesvari, 2006). The strongest Go program as of 2009 is Golly and Silver's
MoGo (Wang and Golly, 2007; Gelly and Silver, 2008). In August 2008, MoGo scored
a surprising win against top professional Myungwan Kim, albeit with MoGo
receiving a handicap of nine stones (about the equivalent of a queen handicap in
chess). Kim estimated MOGO's strength at 2-3 dan, the low end of advanced
amateur. For this match, MoGo was run on an 800-processor 15 terailop
supercomputer (1000 limes Deep Blue). A few weeks later, MoGo, with only a
five-stone handicap, won against a 6-dan professional. In the 9 x 9 form of Go,
MoGo is at approximately the 1-dan professional level. Rapid advances are likely
as experimentation continues with new forms of Monte Carlo search. The Computer
Go Newsletter, published by the Computer Go Association, describes current
developments.
There's no word about how Go is qualitatively different to other games, but maybe the referenced sources say something along those lines. Personally, I took a Masters course in AI two years ago, before AlphaGo and I remember one professor saying that the last holdout where humans can still beat computers in board games was GO, but I don't quite remember him saying anything about qualititative difference. Still, I can recall hearing about the idea that Go needs intuition or something like that, except I've no idea where I've heard that. I guess it might come from the popular press.
I guess this will sound a bit like the perenial excuse that "if it works, it's not AI" but my opinion about Go is that humans just weren't that good at it, after all. We may have thought that we have something special that makes us particularly good at Go, better than machines- but AlphaGo[Zero] has shown that, in the end, we just have no idea what it means to be really good at it (which, btw, is a damn good explanation of why it took us so long to make AI to beat us at it).
That, to my mind, is a much bigger and much more useful achievement than making a good AI game player. We can learn something from an insight into what we are capable of.