We know physical quantities as the result of measurements.
Measurement in quantum mechanics is really very complicated. So, let us restrict to classical physics.
In classical physics, measurements come in two basic forms.
The easy one is just counting. You count some unit. The units do not have to be identical. The count is understood to include this variation in the reporting of the quantity. So, twelve cookies includes the variation in cookies. And so on.
In effect, counting establishes an integer ratio between a standard item (possibly somewhat sloppily designated) and a group of items that are considered similar.
The harder one is measurement of some quantity that has a dimension. Length, for example. Measurement has these parts. A standard is established. Then a ratio is established between the standard and the measured quantity. The quantity is the ratio, and the unit of the quantity is the standard.
Previously, standards were established less formally. Such things as the length of the King's foot was "a foot." Or the length of your forearm was "1 qubit." Lately we have established more and more formal standards, and these have become more and more repeatable.
Some easy examples: (These are a bit outdated because the current standards are based on modern physics.)
- The standard is the Standard Meter held in the standards place in Paris, France. A measurement consists in establishing that a distance is so-many times as long as the standard meter, giving so-many meters as the quantity. This relationship can be established through a number of methods from simple things like putting a meter stick beside the thing being measured. Or laser range finders, etc.
- The standard is the Standard Kilogram. The rest is similar to distance.
- The standard second is a certain fraction of a particular year. The rest goes similarly to distance.
- For other quantities such as temperature, power, density, etc., compound standards are established based on simple standards. Density is mass per volume, volume is distance cubed. Mass and distance are simple. Then the rest is similar.
Some other quantities can be dealt with in this fashion. Color, for example, can be dealt with as a wavelength of light, a distance. Color perception is quite different and depends on many factors besides wavelength. Loudness can be done as energy in the sound. Perception of sound is likewise very complicated.
Qualities or characteristics are not automatically able to be expressed as quantity. "Happy" as a characteristic is not able to be straightforwardly converted to a number. It is possible to do surveys and attempt to coarsely give numerical values. This gets into a related but distinct idea, that of affine characteristics. These are things that can be established as larger or smaller, but not to have numerical values. One thus gets "affine time" where you can determine before and after, longer and shorter, overlapping or not overlapping, and similar things. But not actual numerical values of duration. This is what you experience in the absence of any sort of thing to treat as a clock. A clock would act as a standard, and the numerical value would be the ratio between the clock's standard and the measured duration.
So to summarize: physical quantity consists of a measured amount, and this means determining a relationship to some sort of standard.