No.
Identifying/non-identifying relates to child keys. In an identifying relationship, the child record can only exist if the parent record exists. It identifies the child record. An identifying relationship is mandatory in the child table - it forms (all or part of) the key for the child table.
A non-identifying relationship can still be mandatory or optional, it's just that the child record can exist independently from the foreign key.
Mandatory/optional just tells you whether or not the field must have a value. This might be a foreign key field, but it doesn't have to be. If it's mandatory but not a foreign key, you can put any value you wish into the column (as long as it meets the other constraints); it just can't be null
.