I feel like you're almost talking about how similarity criteria are used in engineering. In the case of elastic deformation, the Poisson ratio comes to mind. Or consider such proposals for chemistry as this: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8607974/
With regard to your remarks on what you're calling magnitude dimensions, a given characteristic might map to a number of them. But if our state of knowledge hasn't found something as rigorous as dimensions, we'd have to enumerate and map out the CCD's some other way.
One type of situation that comes to mind is how sometimes not having a certain characteristic counts against the possibility of it being an A much more than having the characteristic provides strong evidence for the possibility of it being an A. That sort of asymmetry could complicate the use of a rigorous definition that you already trust.
Also, while a tentative definition can sometimes help, I can think of two major classes of situations where a definition could be misleading:
1. There can be situations where it is premature to try to articulate a definition. Consider that before we learned that the number of protons defined the elements and constrained oxidation numbers, elements were given lengthy descriptions and were associated with specific chemical tests to determine composition of materials.
2. There are perfectly decent definitions which do not explain a number of characteristics actually used for the purpose of classification. Consider "heat flow". Or consider the use of equations serving as similarity criteria for comparing physical systems.