Simple, because even in 2023, unless you're basically inside your local internet exchange, the pigeon is faster for bulk data transfer.
Raspberry PI fanatic Jeff Geerling decided to race his gigabit internet connection against some pigeons carrying USB thumbdrives... the pigeons won - and by a reasonable amount too! Well, provided you didn't need to send the data more than a good days drive away... (you'll have to watch the video to find out where Pijeff fits in)
Here's the speed/distance plot he came up with:
Edit: As for why no-one in our world uses pigeons, that's simply because light-weight, high-capacity data storage came a bit late to the party relative to international air freight. However, that's not to say that the concept of strapping some kind of data storage device to some kind of moving object (animal or otherwise) isn't still alive and well.
It's quite common for large science facilities to simply post each other boxes of drives when they need to share a couple petabytes with one another. In fact, this is so common that a number of commercial solutions already exist like Amazon's "snow mobile" (which is literally just a truck carrying 100'000 terabytes worth of drives) - for you know, when you need to upload the entire library of congress by next week...
So TLDR; Pigeons are much faster than the internet over short distances, we just don't use them ourselves because parcel post is easier. However, if your world has high-capacity storage but slow networks and snail mail or if you're in a dictatorship or war zone and thus can't trust the network or the post then it just might make sense to dust off your old copies of RFC-1149 and RFC-2549.
Plus, as RFC-2549 puts it: "One major benefit to using Avian Carriers is that this is the only networking technology that earns frequent flyer miles"