Concept of Equality and Similarity in Complex Data

When something, some object or concept, is comprised of tens or hundreds of dimensions or factors, then how do you compare two or more of such objects without loosing significant information?

Multidimensional scaling?
Dimensional Folding?
Cluster Analysis?
Factors Analysis?
Etc, Etc....

I have been intrigued by this question for many years.
To me this issue seems to share a connection with the issue of hash key generation because both can involve representing one thing with many parts with another thing that has one or a small fixed set of parts.

Finding a series of mathematical or other operations that yield unique values such that when a difference is computed between those unique values, that difference itself is also unique to the inputs.