When I spoke with someone at Bowker about this, they said an ebook was an ebook was an ebook, regardless if epub or mobi and didn't require a different ISBN for what was basically the same version.
Without knowing the exact verbiage of your initial question, I would worry about the accuracy of this statement. They probably meant it doesn't matter which specific version you were selling because they are both ebooks. Not that you can SIMULTANEOUSLY distribute an epub AND a mobi through the same channel with one ISBN. That doesn't make sense based on how wholesale distribution works.
Let's backtrack for a moment and remember the purpose of an ISBN. The ISBN is an identifier used by retailers to make sure they are ordering the correct version of a book. It is essentially a "social security number" for a specific format, so that the retailer knows they are getting the correct version. This is particularly important in brick and mortar. If you think you are ordering the hardcover edition of a book, you don't want a shipment of mass market paperbacks showing up.
But it is also important with ebooks. Very few retailers use the mobi format, because it is an Amazon proprietary format only readable on the Kindle or through the Kindle app. The Apple store doesn't want mobi ebooks. They want epub.
Now the entire conversation is merely academic if you are using Draft2Digital or Smashwords to distribute, because they only distribute the ebook versions. Ingram also only distributes epub. So if you use the ISBN for your mobi version on Amazon and then use the ISBN with D2D or Smashwords or Ingram to distribute your epub, nobody is going to know.
But let's say you are working with a third-party wholesaler (not just an Aggregator). That wholesaler is going to "stock" your different ebook formats in the event a company wants to sell a specific format. So you will have your book listed multiple times in their system under the different formats. Just like a print book wholesaler would have the hardcover, trade paperback, and mass market paperback versions all listed separately.
If they are catering to Amazon customers, for example, they may only want mobi versions of the book. But most companies are going to want the epub, because epub is the current standard for ebooks. By having the same ISBN for both versions, you create the risk of the retailer getting the wrong version of the ebook.