FWIW, I can share my anecdotal periodic itching story:

Somewhat infrequently (a few times every year, for the past 11 years) I will go into a "mode" where showering/bathing of any kind will invariably be followed by itching afterwards. This lasts anywhere from a few days (no scratching) to a couple of weeks (some scratching). The itch is concentrated on my front body and legs (never face/arms, rarely back). Nothing helps much except not scratching while it runs its course.

What I've found interesting is that this doesn't manifest during particular stress / downs / whatever. It can be during an awesome holiday in the summer, or over Christmas, or an ordinary work week. But the itching comes after every shower, as if my body picks up some irritant from the water that doesn't affect my hands/face at all. It also isn't "something in the water where I live", as this has manifested in various parts of the UK and Greece.

On a scale of 0 to 10, I'm at 2 as to how manic I've gone about this: when it manifests, I'll start using hypo-allergenic shower stuff, paste myself with moisturising cream and just wait it out. Dermatologists have been the expected/reputed amount of helpful: "sounds like you're sensitive to something in the water", so they gave me a stronger cream, which also didn't reduce the itching. What worked most was not picking/scratching, or at least not enough to break skin/cause small wounds, etc. Depending on your "itch tolerance"[1] takes a little bit of mental focus, so I wouldn't, say, shower before going to bed, but showering and going out was easy.[2] Then, a few days later, I shower -> all good.

I guess my point is, while mine could also be somewhat psychologically induced, I don't think this is the case. Something seems to trigger an "itch flare" which (a) isn't linked to my mood or stress levels, (b) always goes away if I can "forcefully" ignore the urge to scratch and (c) is somehow water related, but not to a particular water source. I feel like If I gave in to the scratching urges, it would be a whole different kind of rabbit hole. So my issue doesn't really compare in the least with what these folks are going through - just a periodic experience of a similar thing at a lower intensity level. You can't always "push aside" pain, so it stands to reason that you can't always "push aside" the scratching urge, and that's when it takes up a life of its own. I am very grateful for not having to deal with this full-time. Not scratching while itching can be proper work, depending on the intensity/duration of the itching spell(s).

As for what mine is, I have no clue, and as long as we stay on our current terms, I'm somewhat content with that. Allergies and sensitivities can be pretty hard to pin down.

PS: Another reason I don't think my itching is psychological - I just read the article, then wrote this, and I'm not itching - yet ;)

[1] if "pain tolerance" is a thing, and itching has its own receptors and all, ...

[2] Except once in a while, in public/with friends/with people anyhow, the itch will "win" and I'll forget that I shouldn't do my instead-of-itching-last-measure, which is to hit/punch the itching spot rather than scratch (my hand is already moving of its own accord, so something needs to happen). Not a masochist in the least, but rather I've noticed that the little physical pain takes away the edge of the itching. YMMV, not offering this as advice. However, you do, you may want to avoid doing that in public, or at least don't follow up any questions with a grunt like "it was that or flaying myself with this steak knife", or you may stop being invited to dinner parties.