At a guess, Archons are immune to both abilities as being purely mental beings. They are not mechanical (unlike Reavers and Probes), but they are not biological anymore either.
Here's what Blizzard has to say (Starcraft II but should be the same as the original in this aspect):
Although their abilities are already impressive, High Templar and Dark Templar can wield even greater might by laying down their lives to merge their souls and form Archons. These unified Protoss spirits radiate incalculable power, unleashing psionic barrages against cowering enemy forces both in the air and on the ground. With the reunification of the Protoss race, any pair of High Templar or Dark Templar, or even one of each, can be fused together. The Protoss are loath to sacrifice their loyal warriors, but those who make this final commitment are forever honored in the memories of their people.
In times of great need, two High Templar or Dark Templar can merge into a single entity that radiates overwhelming psionic power and rage. The decision to lay down the lives of valuable Templar is never made lightly. Perhaps their energy was spent. Maybe the warriors themselves chose this final act of sacrifice over succumbing to their injuries. Regardless of its origins, the Archon’s presence on the battlefield fills the minds of all beings—psionic or otherwise—with primal fear and awe.
Note that they are described as merged souls, "Protoss spirits", formed by laying down the lives of two warriors. They are certainly not mechanical. They form by two warriors merging on the battlefield. They are not built (as Reavers and Probes are). But it seems that the merging process only involves their souls/spirits. It leaves the biological bodies behind. They end up as pure psionic creatures. As such, there is nothing for the broodling to infest.
I believe that Archons and Dark Archons are unique in this aspect. Every other Protoss unit is either mechanical, biological, or both (e.g. Dragoons).
The reason why it doesn't work is just being reported incorrectly. The Archon is a non-robotic ground unit. It just doesn't have a biological component that is vulnerable to broodlings.
I find this explanation more consistent than trying to shoehorn the Archon into being a non-mechanical robot, even though it directly contradicts the error message. The message is simply wrong.