I was expected to use the Fresenius hemodialysis series bibag conductivity sensor to measure the conductivity of water, but I didn't find any reference about using this sensor.

In the test, the sensor is connected to the Analog Device EVAL-CN0359-EB1Z as a display of the conductivity. I've tried various pin configurations, but I'm still not sure because the conductivity results obtained from both are very different from testing using a probe-type conductivity sensor.

What is the correct pin configuration of the sensor?

 

The sensor and the probe will each require calibration. Since you don’t know the expected output of this sensor if you can’t find technical data you will need to characterize the output of the sensor with reference to a known good measure of the same solution at several points and see how it tracks, then derive the proper calibration from that.

What are you using this for? This sensor has an extremely specific application and may not be suitable for anything else. It is very unpleasant to try to help you working blind, please describe the project you are working on.