No matter your point of view from pictures, you don’t know for sure without testing. A certified electrician should be the ONLY answer.
I thought the same thing at first, but after doing some reading it turns out the GFCI measures the difference between line and neutral, and will work, i.e. protect you from shocks, even without a ground wire. They are not surge protectors that shunt to ground, which is what I suspect the hair dryers were hoping for.
Agreed about calling an electrician though; there are too many unknowns/issues here.
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Especially seeing this all in a wet room turns the whole installation into a real horror show! 
Really @KruseLuds, do yourself and your loved ones a favor and consult a certified electrician 
There has to be a line and a neutral for ANY circuit to be complete.
I had to go back to my books, but you are correct. A ground wire is not required, but the receptacle must be marked “No Equipment Ground”. But NEC Article 250 regarding grounding states that a ground wire is required. This is the code that my installations followed.
In a normal safe circuit, the current in the line and the neutral will be equal. If there is a leakage to ground, then the current flows won’t be equal. Thus, a ground fault.
Now that I had to refresh my memory of Article 250, there is a solution to the OP’s question.
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There is a Shelly +1 relay in the related light fixture which is the other end of the Romex, it is set up correctly and is working properly with a neutral. It sometimes loses connectivity to the network because it is in a thick metal casing inside of the fixture which hampers the WiFi signal. Therefore I decided to remove the Shelly and replace the switch with a smart 4-wire switch which will connect to HA via matter. I will check the wiring at the fixture end to check; if the neutral on that end is connected to one of those Romex leads, then I can use the same lead at the other end of the Romex as the neutral. The previous electrician had essentially done the same with the ground wire. Thoughts?
I give up. I won’t play 20-questions.
Call an electrician.
Nobody else has any thoughts?
I have never seen one like that.
They usually measure imbalance of current on L and N. In practice L and N are both passing through current transformer and if sum doesn’t make ~zero (<30mA), it triggers “relay”. Earth ground is not connected anywhere on CFCI.
If it was detecting current in ground wire, it would not detect current that flows directly to real ground (like human touching hot circuit).
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Thanks for the info. I’ve probably installed upwards of a hundred GFI outlets over the past half-century, and since what I was taught as an apprentice electrician made sense, I never felt the need to research it further. The journeyman’s exam was mostly memorizing the NEC.
I checked, there is no neutral available there, I will either find a switch that doesn’t require one or have a neutral installed…