You can do this by duplicating your family tree. When you import a new family tree, you are presented with a list of the people in the family tree and asked which one you are. I don't think there is another way to do this. 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, and that is the only format I tested this with. 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.
Now you will be asked who you are. Pick the person that you want to go to when you click on "Back to me". Then click "Finish import".
You might want to go back to your account and rename this family tree that you just created, so you can find the one you want later.