The firts code call a template register.php this file needs to be in the root of your theme, if not are defined a subdirectory for template parts like /parts /addons /includes /templates /loops , etc. The way to call this shortcode is [my_Bp_registration_form] if you change this to ‘register’ in the add shortcode line, then you can call it [register]. example
function my_BP_registration_form_shortcode( $atts ) {
ob_start();
get_template_part( 'register' );
return ob_get_clean();
}
add_shortcode( 'register', 'my_BP_registration_form_shortcode');
While learning how to use shortcodes try to use less lines and names shorter
function register_form( $atts ) {
ob_start();
get_template_part( 'register_form' );
return ob_get_clean();
}
add_shortcode( 'register_form', 'register_form');
Note that almost has the same name, so on leaning you have less chance to get a mistake; just avoid to use names that your theme has already active for functions. Also your file can be renamed to register_form.php