At the most fundamental level, measurability among humans is embedded in the concept of counting. Most likely, this measure had let our early ancestors determine and compare the number of people in a tribe or family and other objects vital for survival. Research shows that many animal can count which has been linked to survival skills. Further extending the domain into plant kingdom reveals builtin mechanisms to measure light, and temperature, other factors relevant to survival of plants. For example, a seed germination is triggered by certain measurements in the surrounding environment. This probably can be further argued for to inorganic materials such as formation of ice crystals. Even if the ice crystals are considered to be entering an iffy area, it is quite clear that living organisms have an ability to associate measure to physical quantities that can be sensed. But are things measurable irrespective of sensing?
What exactly makes something measurable? Consider the university recreation center where there is a wide range of exercise equipment. While the facility is open during the day, many students use the exercise equipment. To enter the facility, they use their student ID at the front desk. At the front desk and during the workout, there is often exchange of niceties, courtesy, and smiles. There are calories burnt. During the workout, students use towels and drink water from the water fountains. They also drop the equipment on the ground, that produces shockwaves in the building structure. There are many things in this paragraph that one can be considered measurable. If you were asked to identify measurable quantities, a quick and confident response would be number of towels, volume of water, energy of shockwaves, calories burnt etc. Nevertheless, the feeling happy or courtesy afforded will be treated as sensible but not measurable. Why some quantities are measurable and other not?
Consider laundry staff who wash the towels in a washing machine and need to add detergent to water based on the quantity of towels. The most basic and straight forward measure will be "counting" which will provide a single number reflecting the number of dirty towels produced by the recreation center during the day. Count could easily be replaced with weight or volume and would work equally well. On the other extreme, consider the university staff trying to determine if students are happy with their activity. This technically would be feel-good-happiness produced by the recreation center during the day. Clearly towels are measurable but feel-good-happiness is not. If we placed a human observer next to each exercise unit, they will be able to sense the usage of a towel as well as feeling of happiness. At the end of the day, they will provide a "measure" for towels but only a "sense" of happiness.
What makes towels measurable and happiness immeasureable? I think the underlaying process includes sensing along with adding. Sensing is to identify presence or absence. Adding is the ability to combine multiple presences. In case of towels, the laundry staff can identify if there are towels or no towels. And if towels are arriving in bags from various parts of the facility, they can add the towels. In case of happiness, an observer may be able to tell something about it from facial expressions but cannot report at the end of the day how much happiness was produced. It would be very difficult to add the happiness of two different people who come from a different cultural background or had a different ongoing circumstances in life.
It shouldn't be hard to convince that if we move from psychology into neurology, a specialized instrument can quantify the amounts of serotonin produced. Thus, we can report serotonin produced by the facility during the day. Does this mean we have been able to measure happiness? The answer would be affirmative if the all brains followed the same rule to relate happiness to the amount of serotonin. I think its not the case which makes happiness hard to measure. The same applies to other feelings such as love and hate.
Thus, how can we determine if something is measurable? In my opinion, there are two criterion that need to be satisfied. First, is there a notion of nothingness? Second, is there a notion of superposition? Suppose that "something" of interest is towels-used or feeling-happy. The first criterion is met for both because it is only based on the ability to sense presence or absence. Clearly, absence is the notion of nothing. But second criterion provides a stricter notion of measurability. There is a notion of combining all towels by adding but no such notion exist for happinesses. Thus, what makes something measurable is a notion of absence-presence and a notion of combining multiple instances into a single instance i.e. additivity.
