The definition of "physical quantity" depends on the context being used.
For classical physics, it is something that can be measured. And this requires a definition of "measure." To measure a thing is to establish a relationship with a measuring standard.
In previous eras, this standard might be one's own body. Or the body of some well known person, for example the king. This is probably the source of such standards as "one foot." One could establish the relationship by pacing-off a distance and counting the number of lengths of a foot in a distance to be measured. It would not be particularly accurate, and would be different for different foot sizes.
Such standards were formalized into, for example, the metric system. The standard meter and the standard kilogram and various other standards were created. The purpose for these was to establish a means by which different observers could agree on what the standard was, and so agree on what the measured value was. Modern standards are based on our understanding of such things as relativity. We thus have a standard second defined in terms of vibrations of a particular atom and the standard meter defined as the distance light travels in a certain time.
To establish the measurement relationship, we take the physical thing to be measured and find a means by which we can show it to be a multiple of the standard. To measure mass, for example, we establish that the test object is X times the mass of the standard kilogram. X can be any non-negative real number. There are a number of methods to accomplish this. For example, a balance scale can be used, based on the observation that balance beams follow the law of the lever. So if our test object balances at 1.7 times as far from the fulcrum as the standard kilogram, we conclude our test object is 1.7 kilograms.
Similar processes will apply for other standards. The standard of length will allow measurement of distances. And so on. In some cases we will use combinations of standards to establish combination physical measurements. Speed, for example, requires measuring a distance and a time. In each case we will establish a relationship with the measurement standard.
Two important features of physical quantities of this type are the following.
- They have physical units. These are established by the standard. Measurements of mass are in the unit of the standard, kilograms in the example.
- They are not absolutely accurate. That is, they are finitely accurate and have some degree of uncertainty.
An overlapping means of measurement is counting. This applies to collections of items that are similar in some way that is considered, in context, more important than any differences.
A common situation where this applies is counting oscillations in some such thing as a rotating wheel. If we assume a wheel is turning at a constant rate it makes sense to count the oscillations to determine the frequency. The time part of this measurement still gets measured as before, by establishing a relationship to a time standard. The oscillations are just counted. The sources of error here are the usual for the standard. Plus a new type with the possibility of being inaccurate in counting. The units for such are just "per second." (Or per unit time of your choice.)
Another case where counting applies is for items that are sufficiently standard that, for some purpose, we neglect the differences. We may buy bolts of some standard type by counting them. They will not necessarily be similar enough to be indistinguishable. But for the purposes of a building project they may be similar enough. Thus counting still may have some uncertainty associated. And the unit will be "one bolt" with uncertainty indicated by the variation in bolt mass, size, etc.
The meaning of "physical quantity" in quantum mechanics is hugely complicated and technical. And it generates discussion that can get warm. And it's quite annoying to try to explain it without using equations, which are $not enabled$ in this stack. Let me explicitly dodge doing that part.