Is this the same thing as regular exposure to small bits of a food keeping away catastrophic allergies, like with peanuts? Or is that a different thing?
No. The principle of hormesis is that the body's defensive response to a physiological insult is so sufficiently excessive that it produces a net positive to health. This would occur in response to something ordinarily harmful (and thus stimulating such a response), but at a dose low enough that its harm is insufficient to create a net negative. It's important to note that this is actually a characteristic of a dose-response relationship: anything more than a very slight dose is expected to still be harmful.
Most examples of hormesis, well, aren't examples of hormesis. e.g., a quick glance at Google shows "exercise" being given as an example. Exercise, traditionally, is not something considered harmful at all but the tiniest doses.
The peanut thing is just about ensuring that common antigens are presented while the body is still undergoing the creation of peripheral tolerance to foreign antigens. That is, while the body is still young and learning what is and isn't worth upsetting the immune system over, you should go ahead and introduce it to lots of things you don't want it to freak out on.
