All combined disciplines with psychology bring psychology into these exact or hard sciences, whereas these sciences didn't provide enough to psychology to help it becoming measurable.
Please see the attached article (by physicists) shedding some skepticism on the notion that even the hard sciences have any lock on measurement.
Measurement, plain and simple, perturbs the system you want to know about and collapses some-dimensional information into whatever-dimensional information you ask about. Measurement is necessarily a distortion of the "truth" for its distillation into an explanatory model. No science has it "right" or "exact." This argument is not for psychology to defend itself even as a young science. I actually don't think psychology is all that young, and the so-called "hard" sciences often grew out of or owe great debts to earlier questions and investigations about awareness and perception.
This argument about the relative exactitude or measurability of one or another field is a "red herring," distracting from the real work of science, no matter the discipline, and propagating unhelpful distinctions among scientific disciplines.
Psychological science has been taking measurements since the 19th century. We measure behavioral choices, success/failure on tasks, perceptions, reaction times, self-reported mental states, and many other variables. We also study measurement itself, because the variables we're interested in are difficult ones. So I'm not quite sure what you are asking.
All the variables are difficult ones. There are no simple measurements, there are only measuring systems with sufficiently ornate bias to collapse information into tiny enough packets. Please see the attached for how "velocity," that tried-and-true touchstone measurement of introductory high-school physics, is complex enough to spawn whole careers and fields.
I do thank you dear professors for your clarifications and will come back with a reply shortly with more precisions about reasons that pushed me to ask quastions about degree of contribution of other sciences to psychology.