Appending Multiple Fields In Twig
When using {{ content.field_author_name.0 }} it works because field is rendered with the twig.
The ~ operator converts all operands into strings and concatenates them. Given that content.field_author_name.0 is a render array you need point to the string value of this field. To do this you can inspect it with the devel or twig_vardumper module using dump(content.field_author_name.0). More about debugging twig templates: https://www.drupal.org/docs/8/theming/twig/debugging-twig-templates#s-viewing-variables

Assuming that field_author_name is simple text field with the Plain text formatter, this should work:

{{ node.changed.value|date('m.d.y') ~ ' | By, ' ~ content.field_author_name.0['#context'].value }}


To display node creator name you can use this:

{{ node.changed.value|date('m.d.y') ~ ' | By, ' ~ node.uid.entity.name.value }}