I am not clear on how to use the "Set measurement scale" function.
I set the value of 250 as pixel length and 150.5 as logical length, unit of measurement millimeters.
but when I go to measure with the ruler the measurements do not match.
Your screenshot shows 252 pixels not 250 but, that aside, I've just tested the measurement ruler set to 250 pixels and 150.5mm as you describe. I made sure that 'Use Measurement Scale' was set in the options bar and drew out a ruler line of 250 pixels. The reading showed 150.5 as expected.
This is Photoshop v 26.5 on Windows 11
Dave
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