@Rainbow sent me a treemail last night that read:
"Plumeria is related to Adenium. I have observed aerial roots on 3 of my adeniums, though not as many roots as you have on your plumera. There was some rotting going at or near the base or in the caudex due to overly wet soil in their pots, and I believe the plants were stressed and its survival mechanism caused the plant to produce roots. For example, if the plant did collapse due to rot, the roots were ready to root into the soil it might have contact with."
As soon as it was light enough to get outside, I moved one of the pots up to get better photos and it started raining again. I plan to pull this plumeria out of its pot to check for rotting. And thanks, Rainbow, for another suggestion of why these have these crazy roots!