This description lies somewhere between the two extremes of a theory and mysticism, depending on how amiable one is to the concept. David Deutsch is vocal proponent of the former, Lee Smolin of the latter (he categorizes it as "Mystical Realism").
The general idea was initiated by one of John Wheeler's PhD students, Hugh Everett III, in his 1957 doctoral thesis, which introduced relative state functions and provided the mathematical foundation for what is commonly known as the many-worlds interpretation (MWI).
In The Beginning of Infinity David Deutsch defines quantum computation as "Computation in which the flow of information is not confined to a single history." This definition is consistent with his expressed belief that MWI is a testable theory and the only theory with any power to explain the operation of quantum computers (here - note that Deutsch takes issue with the label MWI).
Deutsch is highly regarded and was the first to explicitly describe a universal quantum computer (ibid.). However,
MWI is a minority view, and many other thought leaders disagree with his stance in this regard (see, e.g., Peter Shor's comment to Mark S's answer below). Another notable thinker, Richard Feynman, commented with regard to MWI, "It's possible, but I'm not very happy with it" (here).
To answer your question explicitly, it's not clear whether or not this is an accurate description.
At a fundamental level, Everett describes the situation in his thesis:
...from the standpoint of our theory, it is not so much the
system which is affected by an observation as the observer, who
becomes correlated to the system.
Feynman expanded on this point of view (here),
...that many-world picture says that the wave function $\psi$ is what's
real, and damn the torpedos if there are so many variables, $N^R$. All
these different worlds and every arrangement of configurations are all
there just like our arrangement of configurations, we just happen to
be sitting in this one.
Deutsch has further refined the concept considerably over the years in both scientific papers (e.g., early: 1, 2; recent: 3) and popular science books (4, 5). He generally speaks of an infinite variety of universes within the multiverse, some proportion of which align with each other in particular instances.
In that sense, your second statement is closer to MWI. From what I understand, I think it would be more accurate to say the universes "doing the computation" were identical at the point of state preparation and branch on measurement.
Contact in the form of "message sending" between universes is prohibited by special relativity. As stated by Everett (page 98-99 of his thesis)
Only the totality of these observer states, with their diverse
knowledge, contains complete information about the original
object-system state - but there is no possible communication between
the observers described by these separate states.
If I understand Deutsch correctly, there is a possibility of some form of directed interference that would allow "an observer to 'feel' himself split into two branches" (experiment proposed here), but message sending between the two branches is still prohibited. Apparently, classification of MWI as interpretation or theory (under conventional scientific methodology) depends largely on the viability of this experiment, or one very similar.
Edit: Revised after reading Everett and Deutsch more carefully.