Custom logo manage by customizer and theme options

Have a look into set_theme_mod():

Creates or updates a modification setting for the current theme

and get_theme_mod():

Retrieves a modification setting for the current theme

With this you should be able to get the logo:

$custom_logo_id = get_theme_mod( 'custom_logo' );
$logo = wp_get_attachment_image( $custom_logo_id, 'full' );

This would set the attachment 12 as the logo

$attachment_id = 12;
set_theme_mod( 'custom_logo', $attachment_id );

Hope, this helps 🙂