Summary: The test configuration seems invalid, leading to unexpected measurements.
The topic of the transformer frame (core) not being grounded was raised in earlier comments, and that is likely to be the cause of your incorrect measurements.
Details: What you have there is a neon sign transformer (NST). The label seems to show that it's a "Franceformer 5020 SE", rated output 5000 V and 20 mA. (I would be interested what the caution label says on the top.)
I could not find that as a current model on the Franceformer (now part of SFEG) website. The closest I could find (visually and specifications) now listed on their website, is the 3020 SEG-U (datasheet).
I'll start with a warning and disclaimer: The voltage at both the mains input and HV output of the transformer (5 kV in your case) can cause death, injury, fire or other damage. Anything you do is totally your responsibility.
I have some experience with NSTs from long ago. To cut straight to the point, you ask in a comment on another answer:
Which picture are you [...] saying is the correct way to read secondary voltage? Left or Right? Because Right seems wrong, but that's giving me a correct measurement, and that's how I was taught to use a HV probe.
The response is: Neither of them is completely correct, and that is why your results are confusing - your test configuration is invalid.
Your NST has only one HV output wire (this type of NST is variously called unbalanced / single-ended / single-end grounded / one-end referenced / one-end grounded). That HV output wire is where your photos show the HV test probe measurement "point", so that is correct.
However on this type of NST, the output voltage is referenced to ground (as you are trying to measure in your left photo) but you have not grounded the transformer. The other end of the HV secondary is internally connected to the transformer frame (core). That is where your other output wire is already connected, as we see in your photos, and the frame should be grounded - usually via the mains earth wire.
See below (from Questions and Answers on Neon Transformers):
You seem to be measuring the correct voltage in your right-hand photo because the "common" (a.k.a. ground) clip for your HV measurement probe is connected to the transformer mains neutral input, which should be within a few volts of ground. So by luck, your measurement seems to work. But that configuration, without the transformer being grounded, is still invalid and unsafe.
You need to:
Make the mains connections safe. Your photos seem to show uninsulated twisted wire connections, from the transformer's mains input wires to another cable with a mains plug. These connections could becomes loose, unintentionally touch each other or other objects etc. with dangerous consequences.
I was taught that the mains input to those neon sign transformers is AC polarity-sensitive i.e. the transformer black input wire must go to the mains hot (live), and the transformer white input wire must go to the mains neutral. I don't know the reason for this, but the fact that these transformers use different colors for the mains input wires (white and black), rather than two wires of the same color, is consistent with polarity being important.
The transformer's body itself must be connected to mains ground. Failure to do so can cause the transformer body to reach a dangerous voltage by capacitive coupling, as well as incorrect output voltage.
The image below (from signindustry.com - Essential Tools of the Trade), shows the correct test configuration, similar to your right-hand photo - except your transformer has only 1 HV output wire, not the 2 HV outputs shown in that image (that's a different type of NST). So the other end of the neon sign would be grounded via the wire connected to your transformer's frame, if using your type of transformer:
Then you should be able to measure 5 kV between the HV output wire and the grounded transformer frame.