Is an Archon a robotical, non-mechanical unit?

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.

and later

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.

There seems to be an exception for the Spawn Broodling ability against this type of unit:

Spawn Broodling affects all ground units with the exception of:

• Probes
• Reavers
• Archons
• Dark Archons

The page later goes on and states:

Broodlings are rarely seen in Zerg vs. Protoss since the most expensive Protoss ground units are immune to the spell (i.e. Reavers, Archons).

The same site lists Archons and Dark Archons as Large Ground Units. So it's not that it is not a ground unit, it just has an immunity to the Spawn Broodling ability.

Edit:

The Starcraft Wiki states for Archons:

Archons are neither biological nor mechanical and are therefore immune to a number of abilities, such as Lockdown, Spawn Broodling, and Irradiate.

All that being said, they appear to be immune to both of these abilities that you mentioned.