You build up a classification hierarchy. Every remaining commonality not described in this one is by definition generic.
You could have designed a different scheme, and supported some of the commonalities in your scheme. There is thus some "accidental" genericity. Some of the genericity is however intrinsic: e.g. the one which applies to relationships between classes in your hierarchy.
A degenerate case is this of a classification collapsed to one indiscriminate node: all abstractions are then generic.
Genericity is bound to positivity; classification to discrimination, i.e. negativity. Classification builds up scopes, contexts.
Genericity is "orthogonal" to classification (in an other dimension). This may be understood in two ways: