If I want do do something like this, I do the following:
(1) Decare a parameter in the (generic) part that will hold the value of interest.
(2) In that same part, use a relation to assign the result of the analysis to the parameter.
(3) Now you can access the value in the usual way, by calling it with the appropriate session id of the part in the assembly.
Also, something I see a lot is people creating measurements, sections, etc. in a family table instance -> this results in the created items being suppressed in the generic and all other instances of the part, as well as them all being added to the family table.