Hi COOL,
I'm not sure what the same function means? "would like to have the same functionality in Case's Contact lookup".
So you want to primary contact related to account also can be shown in the contact lookup field of the case entity?
You said Contact & Account is N:1 relationship(Existing contacts sub-grid), More than that, Contact & Account have another relationship is 1:N(Primary contact lookup).

Only child contacts related with the account can be listed in the Case's Contact lookup, the parent contact cannot be listed.
If you want to view primary contact related to the account in case:
One way is adding the primary contact to the sub-grid;
Another way is creating a new 1:N relationship between Contact and Case, add the new contact lookup field to case form, then you can get primary contact from related account to the new lookup field in case.