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So in the latter segment he is not saying that every register can be accessed simultaneously, he is saying that every register has the property that it can be accessed in the same amount of time, regardless of how many registers there are total.
That's actually not quite a true statement, but it's close enough.
He's trying to emphasize the importance of RAM being "random access" memory and I think he could do a better job there (my opinion, for what it's worth). With RAM I can access any memory cell I want to at any time and the cell that I accessed last has (virtually) not effect on how fast I can access any other cell next. This is a big departure from another kind of memory known as "serial" memory in which I can access the next memory cell (in address order) very quickly but it can take a significant amount of time to access a memory cell that is not the next one.
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