In my World of Warcraft directory, there are three folders: classic, classic_beta, and classic_ptr. (I used to have a fourth, retail). The launcher just updated the choice from World of Warcraft Classic to Burning Crusade Classic but I see no change yet in my WoW directory. Each version of WoW had its own Interface/Addons sub folders and each version will require its own addons. You can copy your addons folder to the new one, but you might want to check Curseforge for addons specific to TBCC.
edit: now there are four: classic, classic_beta, classic_era, and classic_ptr
Your Classic WoW should update from 1.13 to 2.5. With that, all your addons and settings moved on to TBC. Classic Vanilla wow would need to be reinstalled. So in short, yes everything moved over to TBC. My Directory updated with the patch and now shows: Classic Classic_era classic_ptr ptr retail
In this case “classic” is TBC while “classic_era” is my new install of 1.13. Classic Era was not there until I reinstalled which is why some of you may not notice a difference in the directory names.