Why is there no Customer Group option when I edit a customer?

The Customer Group drop-down on a customer's Profile tab only appears when the Customer Groups plugin is active on your store. Check that Customers → Customer Groups opens your groups rather than a plugin notice; if not, activate the plugin from Plugins.

If the drop-down is there but your group is missing, the group is disabled. Go to Customers → Customer Groups, click Edit, tick Active and click Save group. Only active groups can be picked.