Both are wrong to me. Make mine tea, please.

Both are correct. You can use "doing" for a lot of activities: I'm doing the coffee now. I'm doing my hair. I'm doing my nails. I'm doing the cleaning. I'm doing accountancy as my major. I'm doing computers at that school. The list is almost endless.

Making because you are creating it.

The person who said both are correct is wrong. "I'm doing my coffee," is not what anyone (i know at least) would say. I'm making my coffee (or probably moreso - I'm making a coffee) I'm fixing myself a coffee (another colloquial way to say you're making yourself a coffee)

I would say: I'm making my coffee.