The MEAN stack, just like LAMP (Linux, Apache, MySQL, and PHP), is simply a stack of different programs: MongoDB, Express, AngularJS, and Node.js. It's more of a guideline—you can definitely use a different database system such as Postgresql, a different webserver in place of Express such as hapi.js, or a different front-end framework such as Ember.js.
The MEAN one-click is basically an Ubuntu Droplet with MongoDB and Node.js preinstalled and a sample MEAN project placed in
/opt/mean, which means that you can manually install these components on a fresh Ubuntu Droplet yourself and get the exact same result.
The one-click is intended to be an image that people can use as a quick start for MEAN projects on which they can build on top.