How can I pass a variable defined in a Controller to my variables?
The shortest answer is "that's not possible," and the slightly longer answer is "that's not possible because Controllers and Variables have no reason to ever interact." If your Controller relies on a Variable (or vice versa), something is architecturally incorrect. There's no reason for the two to cross paths.

The purpose of a Controller is to receive input from a form, AJAX call, or to be used as an API endpoint. It can also be used via routing to show a particular template under particular conditions.

The purpose of a Variable is simply to render a value in your Twig template.