WP Customizer API loaded into functions.php

Yes. You can load the code in the customize_register action. One example:

<?php
// File: functions.php
add_action( 'customize_register', function( $wp_customize ) {
    require_once dirname( __FILE__ ) . '/inc/customize.php';
    wpse256532_customize_register( $wp_customize );
} );

And the inc/customize.php file:

<?php
// File: customize.php
function wpse256532_customize_register( $wp_customize ) {
    $wp_customize->add_setting( /* ... */ );
    $wp_customize->add_control( /* ... */ );
    // ...
}
// ... any additional customizer classes and other includes ...