Is there a simple phrase to describe the object of an agent noun performing an action?
The adviser helped the advisee.
The employer fired the employee.
Is there a phrase similar to "agent noun" that can describe advisee and employee?
Absolutive. But that also includes the subject of intransitive verbs.
Adding the suffix -ee to a verb produces a label for a person who is the Absolutive of the verb – i.e, a person who is either the Direct Object of a Transitive verb, or the Subject of an Intransitive verb.