How to save datetime field value in config?
There is no schema for data type called datetime.

Easiest solution is to store the date as the timestamp string.

Converting a drupal datetime into a timestamp:
The datetime form element holds an array value that contains three keys: date, time, object.
The object holds an instance of the [\Drupal\Component\Datetime\DateTimePlus][1] class. It has a _toString method which means it can be converted to string via normal casting (i.e. (string) $value).

Setting a default value on a datetime field:
Too do that you will need to create a new [\Drupal\Component\Datetime\DateTimePlus][1] object and set it as the value of $element['#default_value']