but I am quite certain (from the last time I read the documentation) that that cache is associated with the package STASH, rather than each object, since it logically applies to the class (package) and is shared between all objects of that class.

That is the case.

$ perl -M5.010 -e'
   package Foo { sub f { } }
   package Bar { our @ISA = "Foo"; }
   say for keys %{ $::{"Bar::"} };
   say "---";
   Bar->f();
   say for keys %{ $::{"Bar::"} };
'
ISA
---
ISA
f
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