The tags component uses JS to add the tags in the frontend.
The JS file is a single PHP file that puts together all the necessary JS files. One of the JS files is a JSON encoded configuration file. The configuration file is not being loaded, or loading of the JS files stops when the configuration file is being encoded into JSON format.
It may be a PHP fatal error and you have error reporting turned OFF in PHP (which is the best setting for production sites).
Could you add the PHP code:
<?php
error_reporting(E_ALL);
?>
To the top of the file components/com_tag/tag.php. This will show what error is occurring.