If I want to implement the relationship between Category and Classified, is a database-level nullable foreign key required or is it possible/advisable for an application to define this type of relationship without using a database constraint?
[Note: Because the white dot indicates “optional” and the black dot “required”, for each Category a Corresponding classified may or may not exist. In addition, the crows feet between them indicate this is a many to many relationship.]

Since it’s a many-to-many relationship, you’ll want a cross-reference table rather than a simple foreign key column.
So the Category table does not have a FK to Classified, and Classified does not have a FK to Category. Instead you can have a new table :
This is a typical way to implement a many-to-many relationship. And now, instead of worrying about NULLable fields if two records aren’t related, you simply care about the existence or non-existence of a xref record