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I consider a "normal", comfortable working distance (when photographing people) to be about 6 ft (180 cm). But then it occurred to me that there might be a technical definition for a "normal" distance because there's a technical definition for a "normal" lens, where the focal length approximately the length of the diagonal of the frame.

Is there a technical definition for "normal" working distances at which to photograph subjects?

I don't think this is a commonly-agreed upon or standard term. The phrase "normal distance" does not appear in the Focal Encyclopedia of Photography, and Google searches for 'photography "normal distance"' do not show any real pattern of use. (Most references are either as opposed to macro distance or actually referring to normal viewing distance for prints or screens.)

Because perspective is solely determined by distance between camera and subject, there's at least a common-sense answer to this: a normal distance is the same distance one might stand from that subject to observe or interact with it in real life. For people, it can range from a conversational distance to, maybe, across the room. Closer than that looks odd, as does an ultra-telephoto perspective. The same can apply to buildings, bugs, cars, or whatever else you're photographing.

Regardless of focal length, portrait perspective is determined only by the distance at which the camera stands, which is important. This could include like not making the nose appear larger than it is. Focal length does of course influence where you need to stand.

It is all opinions, but 6 feet might be considered a minimum. 5 feet can be too close for portraits, and 4 feet absolutely is. Many studios prefer 8 or more feet for formal portraits that they hope to sell. The 105 mm lens was popular for 35 mm film portraits solely because it forced standing back properly. Some preferred a 135 mm lens for same reason (speaking full frame, cropped sensors would use shorter lenses to stand back at same distance).

Focal length and subject distance can vary, depending if it is a head and shoulders shot, or full length or group. But in every case, there is this same minimum distance for proper portrait perspective. Zoom in all you want, but do stand back a bit.

Some users imagine a 50 mm f/1.8 lens is needed for outdoor portraits, thinking to blur the background. The pros would avoid the f/1.8 issues, and would likely use a 200 mm lens to blur the background even better, plus better portrait perspective too.

It relates to binocular vision and stereopsis. Stereopsis is where two distinctly different images exist on the L/R retinas due to angular differences; which the brain then combines as a single image with binocular depth information (as opposed to monocular cues). The brain does some amazing things in this situation in order to create a "normal image" perception.

This varies both by individual and situation (i.e. the distance between near/far objects). But the significant majority of it occurs at less that 10ft... beyond 10ft monocular (2D) cues are much more significant (some say beyond 3-5ft).

So the "normal distance" (perspective) is that where normal monocular/2D vision/cues align with the resulting 2D image... most commonly accepted to be around 10ft when photographing facial features IME.

The premise of your question is flawed in that a normal lens isn't really a technical concept. A normal lens having a focal length approximately equal to the diagonal of the film or sensor is just an observation, not a definition. A normal lens is considered normal because it fills the frame with common subjects at comfortable working distances and natural perspectives. What we consider a comfortable working distance or natural perspective is a function of the distances at which we normally observe things in our everyday experience.