Do one-on-one simul competitions exist? I have occasionally played one-on-one two-board simuls: that is, two players, playing two games of chess, against each other, simultaneously. (Not to be confused with bughouse - no piece passing takes place).
This concept would seem to scale pretty easily with web technology: play any number (well, probably an even number) of games between two people simultaneously, and the score for the match is how many games you win.
You could then have a tournament in this format: each pairing is not a single game, but X simultaneous games (maybe 4 or 8 is a good number).
Has this been done? What is the name for this style of competition?
Yes, one-on-one simultaneous competitions exist. For example, you can find them in correspondence chess. One modality is as two game matches, where you play your opponent with white in one game and with black in the other. Similarly for friendly matches between federations of different countries, with two games for each board.
There are also several websites where you can play simultaneous matches. Have a look at the answers to the question Is there any website where I can play simultaneous chess against many opponents?.