I don't think he means in terms of implementation detail like "how many machines are running the world": the idea is that there is only a single shard (to use UO terminology; I believe WoW calls these "realms"), which means that everyone is playing together in a single massive alternate universe.
I agree, that is probably what he meant, but it was factually incorrect.
Even so, in eve there are only small aspects that span servers simultaneously (the market place, and maybe other things too--I haven't played in a year or so).
The unique thing about eve is being able to jump between "realms" or servers or "shards." That gives the illusion of playing together in a much larger world.
I've never played WoW, so I don't know if you can jump between shards or realms, or if they are treated more like parallel dimensions.
It wasn't factually incorrect, you just didn't understand him because you're not using the video-game-specific meaning of the word "server."
As you suggested, it refers to multiple "parallel dimensions" between which the playerbase is split to avoid overpopulation; that's how almost all other MMOs work. Whether each of those runs on a single physical computer is irrelevant to anyone except the host.
Depends on the definition of jumping. Besides paid character transfer (which is a "true" jump from realm to realm) a lot of activities in WoW take place in "instances" and players from different realms may participate in the same instance (PVP matches, dungeons, raids). A few weeks ago crossrealm zones were introduced, so normal questing areas can now span several realms too.
It's kind of a hybrid between EVE's approach and classic UO's "shards as isolated universes" system.
WoW is not that sophisticated. Or perhaps a better way of thinking of it is that Blizzard wants to provide a highly predictable customer experience. The world has no real state so it's simply like being locked into parallel _identical_ dimensions.