Since you are only updating from TeamSpeak Server v3.13.7 to v3.13.8, there is no special migration procedure required.
Before updating, I recommend creating a complete backup of your current TeamSpeak server directory. This will ensure that your database (ts3server.sqlitedb), uploaded files, icons, configuration files, and any other custom data are preserved.
As long as you keep your existing ts3server.sqlitedb, your channels, ban list, server groups, permissions, server group IDs, rank icons, and other server settings will remain intact after the update.
For the update procedure itself, I would like to refer you to the following guide:
The guide describes the recommended update procedure step by step, including how to properly stop the server, create a backup, replace the server files, and start the updated server again.
If you follow these steps, you should be able to update to v3.13.8 without losing your existing server data.