It would help if we had actual information about the specific mug. I would find it odd that the interior of any mug would be anodized or powder coated and would not want to use either for drinking. You do not want to use either for generating smoke unless maybe you were using a slow smoldering dust or pellet type system, assuming they dont generate much heat.
You can look all this up, but in short, anodizing is a charged particle electrically applied finish. Metal is immersed in a solution with the color or pigment while bieng charged with electricity drawing the color/coating to the metal's surface.
Powdercoatong is a paint coat that is applied to the surface in powder form, then baked in an oven to congeal and adhere the powder/color to the metal's surface.
As a former welder and metal fabricator, I can tell you both are no good to inhale when burned and you cant weld over either finish. The finish must be ground away to successfully weld the metal underneath.
Having not seen the mug you speak of, it is possible that the interior of the mug, where the liquid would be, is stainless, and the exterior anodized or powdercoated. While you could light a fire inside the mug, it may burn off the finish on the outside. Which would not be good to breathe...not in the smoke but just standing around it externally. Also, if its an insulated mug with an air chamber between two layers, if its not vented somewhere it would blow up from pressure buildup. You could avoid that by drilling a hole through the exterior wall to allow it to vent.
