Is it possible to create exclusive custom taxonomy?

The default behaviour of WordPress doesn’t allow this.

So, to do this, I’d suggest the following method:

1. Register the taxonomy as usual
2. Remove the taxonomy meta box so users don’t select the taxonomy term the old way
3. Create your own meta box with your own UI to replace the taxonomy meta box. So users can only select a term from your radio list or select dropdown.

The 2nd step can be done with this code:

add_action( 'add_meta_boxes', function() {
    remove_meta_box( $id, $post_type, $contex'side' );
} );

$id is the ID of the taxonomy meta box, which can be either 'tagsdiv-{$tax-name}' if the taxonomy is not hierarchical (like tags), or '{$tax-name}div' if the taxonomy is hierarchical (like category).

The 3rd step can be done manually, but as it involves with outputting fields, handling sanitization, saving data, I’d suggest using a library like Meta Box plugin to do that. (Disclaimer: I’m the author of the plugin).

The code to do for meta box looks like this:

add_filter( 'rwmb_meta_boxes', function( $meta_boxes ) {
    $meta_boxes[] = array(
        'id' => 'custom-meta-box',
        'name' => 'Taxonomy Name',
        'context' => 'side',
        'fields' => array(
            array(
                'id' => 'custom_taxonomy',
                'name' => 'Taxonomy Name',
                'taxonomy' => $taxonomy_slug,
                'type' => 'taxonomy',
            ),
        ),
    );
    return $meta_boxes;
} );

More docs can be found here.