If Nintendo had a larger stake in Niantic they wouldn't have felt the need to release this PR and this thread wouldn't exist.
Its not the stake in Niantic. Its the stake in this project, and I find 13% to be downright unbelievable and an analyst lowball (do not listen to public analyst opinion). Nintendo has a trademark cut that will generally come off of the top of the earnings along with Google/Apple's 30% cut off of the storefront. There is no way I find "13%" as a believable figure.
They have 33%-base ownership in TPC directly, an unknown ownership in Creatures Inc. (its not 10%, that is a made up analyst guess, it has never been disclosed), and a Trademark ownership cut for the characters and Pokemon property. There is a further a large investment from both Google and Nintendo in Niantic directly, and we have no idea how this is being handled whether its shares of the company (as it is private) or funding directly into the project with kick-backs from Niantic's earnings.
There is next to zero chance that Niantic is somehow making more money than Nintendo in this deal. However, if they follow operation as they normally do with external Pokemon projects, the earnings here will not be directly on their own projections but in equity accounting on the side through TPC. Due to this nature of reporting, TPC gains or Pokemon gains on project not directly tied to the mainline games that Nintendo publishes (which will go back to Nintendo) do not show up on the earning sheet directly under Nintendo's own earnings. If the only thing Nintendo is going to factor in to their direct reporting from this app are a trademark cut, then... ya, its not going to amount to much relative to the rest of their business and isn't going to magically swing their reports to some stratospheric highs.