Is the MODULE.schema.yml required when you define the MODULE.settings.yml? Is the MODULE.schema.yml required when you define the MODULE.settings.yml? It seems that it also works when I remove the MODULE.schema.yml file from install/schema.
It is not absolutely required but strongly recommended to have one.

By default, Drupal doesn't validate the schema, but it does in tests and there is an issue that will enable it if you use the provided example.settings.local.php/development.services.yml. That's why you don't see an error right now when you install, but you'll see it when you write your first test.

It is currently used to ensure the saved configuration has the right data type (e.g. a float and not an integer); not doing that correctly can break stuff and the ability of translating the configuration. More things might follow later, for example a module that generates a settings form.