This isn't true, or at least, it isn't true most relevant ways.
Most obviously, if I need to specify which record in my database (of, say, a million records) is to be read, then "3" and "123321" contain precisely the same amount of information.
But I think that even in more common cases: "how many birds are at the bird feeder?"; "how many gallons did it take to fill you gas tank?"; "what's you GPA?", the answer, regardless of its magnitude, conveys the same amount of information.
Now, precision is a different story. "I'm going to take a couple of weeks vacation in July" contains less information than "I'm going to take 11 days of vacation in July". "My gas tank has a capacity of 16 gallons" contains less information than "my gas tank has a capacity of 15.9 gallons.".
Like in the article it could be far more likely to have a 1 follow a 1 (like markov chaining probability) than another number, making the string A more probable and therefor containing less information.
The overarching notion to me seems to be (Kolmogorov) complexity. The size of the smallest program that can reproduce the string/word/utterance denotes the complexity of that string/word/utterance.
We use less complex words more often, not because they are shorter, but because their optimal compression ratio (not the number of morphenes, but their order and complexity) is shorter than other words. The program to produce it in our brains is smaller and simpler.
Last week, the results were 11111 11111 11111. This week, the results are 48611.
In this case, would you suggest that last week's results convey less information? It does convey overall more facts than this week's results.