What happens in a rare occasion you like to duplicate something at higher octaves? Musical rule says there is no doubling upwards, either going down, or staying where you are! Because there is no term for it?
Re: Doubling = always downward? Whas the opposite?
I think doubling just refers to using the same note of a chord (or even a whole melodic line) on a second instrument/voice.
You are free to double either in unison, or up or down the octave (or even wider!). Doubling doesn't specifically refer to one note/part playing an octave down an octave from another.
Indeed, if two instruments/voices play the same part in octaves, who's to say which is the lead part and which is doubling the other (although it is quite common to enhance a melody part by later adding a 2nd line playing the melody an octave above/below the first, where the answer is clear in such cases).
I found this page on google by searching for "harmony doubling" - it is sometimes difficult to know what to search for in search engines!
Remember that originally there were no musical rules ... these have only evolved over the centuries according to what leading composers/musicologists deemed sounded good or bad.
All that really matters is what sounds good to you, though it is well worth trying to understand what rules have evolved (or are adhered to in a particular style). You are completely free to apply them or ignore them as you wish, though this may affect the popularity of your music with musical "fundamentalists".