Dependency injection not working custom form
I accidentally declared the $custom_manager property of my class as private instead of protected, and that's why I was unable to access the custom manager object in the form submission method. Whoops!

MPD's answer is theoretically correct though, as class requiring to access services do need to implement the ContainerInjectionInterface class. 

Worth noting though that the class FormBase which we're extending, already implements ContainerInjectionInterface, so we don't need to explicitly do so when creating a custom form.