Concept for handling multiple notification templates. We went with a generic template solution, but saving this in case it is needed later.

customize-notification.php

<?php
/**
 * Send a template for handling notifications that other plugins can hook into.
 */
add_action( 'customize_controls_print_footer_scripts', function () {
	?>
	<script type="text/html" id="tmpl-customize-unified-notifications">
		<ul>
			<# _.each( data.notifications, function( notification ) { #>
				<li class="notice notice-{{ notification.type || 'info' }} {{ data.altNotice ? 'notice-alt' : '' }}" data-code="{{ notification.code }}" data-type="{{ notification.type }}">
					<?php do_action( 'customize_unified_notifications' ); ?>
				</li>
			<# } ); #>
		</ul>
	</script>
	<?php
} );

/*
 * Default message at priority 10. Use 9 and 11 to go before or after.
 */
add_action( 'customize_unified_notifications', function() {
	?>
		{{ notification.message || notification.code }}
	<?php
} );


/*
 * Example of usage in customize_concurrency.
 */
add_action( 'customize_unified_notifications', function() {
	?>
	<# if ( /concurrency_conflict/.test( notification.code ) ) { #>
		<button class="button concurrency-conflict concurrency-conflict-accept" type="button"><span class="dashicons dashicons-thumbs-up"></span></button>
		<button class="button concurrency-conflict concurrency-conflict-override" type="button"><span class="dashicons dashicons-thumbs-down"></span></button>
	<# } #>
	<?php
}, 9 );

/*
 * Function to make adding a dashicon simple.
 */
function add_customize_unified_notification_dashicon ( $code, $dashicon ) {
	add_action( 'customize_unified_notifications', function() use ( $code, $dashicon ) {
		?>
		<# if ( /<?php echo sanitize_key( $code ); ?>/.test( notification.code ) ) { #>
			<span class="dashicons dashicons-<?php echo sanitize_key( $dashicon ) ?>"></span>
		<# } #>
		<?php
	}, 9 );
}