There is a standard object ApexEmailNotification
which stores information about Apex Exceptions Email Settings
It is possible to retrieve these data using SOQL or Tooling API SOQL
SELECT CreatedById,CreatedDate,Email,Id,IsDeleted,LastModifiedById,LastModifiedDate,SystemModstamp,UserId FROM ApexEmailNotification

Despite this object is defined as EverCreateable, it is not possible to use Apex to create this records, if you try to
insert new ApexEmailNotification(UserId='0051t000002So5jAAC');
you will receive an error
DML operation Insert not allowed on ApexEmailNotification

Even though DML operation Insert is forbidden on this object, there is still an option to use Tooling API POST method to create this records by sending JSON body to /services/data/v45.0/tooling/sobjects/ApexEmailNotification
endpoint

or even using usual endpoint
/services/data/v45.0/sobjects/ApexEmailNotification

So basically it is possible to write a script in SandboxPostCopy which would query production data and use self callout to insert them in Sandbox after refresh.