There's no Copy button, but you can accomplish the same thing by exporting your family tree and importing it as a new tree. Here are the steps:
To duplicate a family tree, download your family tree in Family Script or GEDCOM format. I would suggest Family Script format because that is the native format for Family Echo. You can download your family tree via the "Download / export this family..." button at the bottom of the left sidebar.
After you download your family tree, go to your account by clicking on "My account" in the upper right. On the account screen, click on "Create or import a new family".
At the bottom of the left sidebar, click on "Import GEDCOM or Family Script...". Browse for your family tree file that you just downloaded and import it. Make sure you save it with a name that will help you differentiate the two family trees.