When copying an existing Cassiopeia child template, Joomla creates an inconsistent media directory.
The copied template directory is created correctly, but the associated media directory is created with a different name.
As a result, the Template Editor displays the message:
The template directory is not writable. Some functions will not work.

The directory permissions are correct. The error is caused by the incorrect media directory name.

Steps to reproduce
Create a Cassiopeia child template (for example cassiopeia_child_altlayout).
Open System → Site Templates.
Select the existing child template.
Click Copy Child Template.
Enter xy as the new template name.
Open the copied template in the Template Editor.
Actual result

The following directories are created:

/templates/cassiopeia_child_xy
/media/templates/site/cassiopeia_xy

The media directory is missing the child part of the template name.

The Template Editor reports:
The template directory is not writable. Some functions will not work.
although the directory permissions are correct.

Expected result
Both directories should use the same generated template name:
/templates/cassiopeia_child_xy
/media/templates/site/cassiopeia_child_xy

Workaround
Rename
/media/templates/site/cassiopeia_xy

to

/media/templates/site/cassiopeia_child_xy

After renaming the directory, the Template Editor works normally.