The MailVerifyPresenter class has the following constant :

private static final String SIMPLE_CONFIGURATION_NODE_PATH = "/modules/mail/config/simpleConfiguration"; 

5.7.x has this path set correctly and can be seen from the Configurations within /modules/mail/config/simpleConfiguration.
(see screenshot)

6.1.x has the location of this configuration under /modules/mail/config/templatesConfiguration/simpleConfiguration.
(see screenshot)  

As a result, the mail module does not work without manually moving the node to /modules/mail/config/simpleConfiguration.

Solution would be to change the path within MailVerifyPresenter, or move the config node to the correct location.