It's necessary, to make the claim that they aren't the same character, which was the context of the post I was quoting. Otherwise, it's unknown.
They're obviously teasing us with the idea that Burnham is at least in some way inspired by the character in "The Cage." Not only is the name "Number One," the sex of the character, and position on her ship all the same, and presumably rank as well, but also Michael Burnham's initials are the same as Majel Barrett's. It could just be homage, of course.
And, yeah,
@Mr. Laser Beam is right about the years. If the years mentioned in the trailer turn about to be right, namely when DSC is set and how long Burnham as been serving with her captain according to how the trailer is cut, it seems super-unlikely now.