He may be saying that sending the message "MY HOUSE IS BIG" twice communicates no more information than sending it once. It's a little difficult to tell since there are so many details he doesn't make explicit.
My problem is that SO seems to make an awful lot of assumptions regarding how to measure information.
I forget the specifics of Shannons theory, since its been, hmm, to many years since university.
But of course context matters. I remember an example with child birth. The father to be is outside the door, and he desperately needs to know the sex of the newborn.
Now The nurse might come out the door and say "It's a boy", a message which is 10 characters long, but they could have agreed on a protocol, such that she just displayed either a black or white piece of paper in the window in the door, white for boy and black for girl.
The information content would be the same for the father. But this is just because the set of possible messages is exactly 2, so either way she could at most communicate 1 bit of information.
Now claiming that
MY HOUSE IS BIG
MY HOUSE IS BIG
has the same information content as
MY HOUSE IS BIG
is just silly, when you do not specify any context. As my simple example showed, repeating the message can communicate more information. And indeed it must, since even if you get no new information about his house when he repeats the statement, you still gain new knowledge, since you now know he said it twice.
Since we are talking about biological entities, we should look at DNA. I know nothing of how it works, beyond a cartoonish impression,
but to take our analogy to the breaking point, will duplicating a gene never ever have any effect on the phenotype of an organism.
Notice however that this a question quite separate from whether there is more information or not.