The measurement result looks plausible to me. A LISN measures the sum of differential and common mode noise. The current probe measures common mode noise, if both supply lines pass through the aperture. At low frequencies, differential mode noise is dominant. As a rule of thumb, common mode noise starts to become dominant at higher frequencies, typically above 10 MHz. On your plot, the spurious at approximately 3 MHz and 6 MHz are visible, which points to common mode noise. The spurious at low frequencies are differential mode and consequently not visible. In order to measure differential mode noise with a current probe, you need to feed the second supply line in opposite direction.
Also ensure that the power supply lines are fed through terminated LISN, to establish a defined source impedance.
For more details, have a look at this excellent article of Kenneth Wyatt:
https://interferencetechnology.com/measuring-common-mode-versus-differential-mode-conducted-emissions/
What does not look plausible to me is that the test-house plot matches the result measured with the LISN, as it contains the differential mode noise.
A quick health check is measuring the DC resistance from the connector center pin to ground. It should be a DC-short. For a full characterisation, a measurement using the calibration fixture is required. You could send the probe to Tekbox to get it checked.