The physical constants are part of the observer's
data- as events are usually first observed happening in nature or in a laboratory set up.
To build a relationship between observed numbers/properties and the varying environs or properties of materials or measured values a constant appears- which may be universal or restricted to the 'situation'.
Many times those constants get a physical definition when detail investigations become available. An example is 'mass' which plays a role of a constant in dynamics of bodies-or spring constant k, which plays a role in defining the energy of a spring { (1/2) .k.x^2 }, where x is the extension of spring.
or Reynold number in the flow of liquids.
If the constants have dimensions it means that more info is required about the phenomena.
not only in physical sciences but in psychology or social sciences such properties may exist which may be related through some constants specific to the environs.
About the nature of physical constants-
Duff M.J. argued that the laws of physics should be independent of one’s choice of units or measuring apparatus.
This is the case if they are framed in terms of dimensionless numbers such as the fine structure constant, α.
For example, the Standard Model of particle physics has 19 such dimensionless parameters whose values all observers can agree on, irrespective of what clock, rulers, scales... they use to measure them.
Dimensional constants, on the other hand, such as ¯h, c, G, e, k. . . , are merely human constructs whose number and values differ from one choice of units to the next.
In this sense, only dimensionless constants are “fundamental”.
Similarly, the possible time variation of dimensionless fundamental “constants” of nature is operationally well-defined and a legitimate subject of physical inquiry.
By contrast, the time variation of dimensional constants such as c or G on which a good many (in my opinion, confusing) papers have been written, is a unit-dependent phenomenon on which different observers might disagree depending on their apparatus.
All these confusions disappear if one asks only unit-independent questions.
ref.https://arxiv.org/pdf/1412.2040.pdf