What is the name of this peppermint stick candy?

Do you mean something like this?

https://www.giambris.com/Porous-Peppermint-Sticks-1-Doz_p_1810.html

Porous seems to be a potential keyword, you might also browse through “old fashioned” candies or candy sticks for familiar brands or descriptions, especially since you do seem to be searching for something previously common but not currently… in fashion i suppose. I remembered this one while looking for a different “old fashioned” candy a while ago and just remembered it when i saw your question.

I also know the difference between peppermint sticks and candy canes (which is the terminology I've always used), but part of the problem is that names vary a lot by region.

Candy canes and peppermint sticks begin similarly: you cook sugar until very hot - at least 270 F, if not more depending on the recipe. The first difference between the two is that peppermint sticks will likely remain on the lower edge, while candy canes will likely go a little over 300 F. The lower the temperature, the softer your final product will be. Peppermint sticks are also liable to contain butter to add to that melt-in-your-mouth texture. Once the sugar syrup has come to temp, you remove from the heat, pour onto a marble slab (or other work surface). From there you can add and stir in flavors and colors.

The second and biggest difference is in how the sugar syrup is worked. For candy canes, the sugar syrup is pulled to develop that white, opaque and shiny characteristic. (Check out Hercules Candy or Lofty Pursuits on YouTube to see it done.) For peppermint sticks, like classic fondant or fudge, the sugar syrup is deliberately worked until they crystallize, becoming opaque and not-shiny.

Candy canes and peppermint sticks are both pulled once the sugar syrup has been appropriately prepared. Candy canes get a crook while sticks are left straight.

I managed to find an example recipefor soft peppermint sticks as well which describes the procedure/appearance, which might be helpful.

That said, a final note is that candy-making in general takes a lot more skill than is often (or easily) communicated, and pulling candy is honestly difficult to do at all, let alone so it looks good. Just be careful with any hot sugar syrup and do plenty of research before you start trying to pull or work anything capable of burning you.