Show all Tags in each post

Considering you mention having registered a custom taxonomy instead of using the native tags, the functions the_tags() and get_tags will not work.

you need to get your custom taxonomy terms by using get_the terms(). Usage is not much different from using native tags, but you have to pass the post_type as a variable. See the Wordpress codex on this: https://developer.wordpress.org/reference/functions/get_the_terms/

Check out get_the_tag_list() to retrieve all the tags:

https://codex.wordpress.org/Function_Reference/get_the_tag_list

Examples:

$separate_meta = __( ', ' );
echo $tags_list = get_the_tag_list( '', $separate_meta );

OR:

<?php
 $tags = get_tags();
 if ($tags) {?>
  <ul class="tags">
  <?php
    foreach ($tags as $tag) {
    echo '<li><a href="' . get_tag_link( $tag->term_id ) . '" 
    title="' . sprintf( __( "View all posts in %s" ), $tag->name ) . '" ' . '>' . $tag->name.'</a></li>';
  }
  ?>
  </ul>
<?php }?>

You have to be careful when creating a custom post type with taxonomy and the same namespace "tags", since tags are already registered by WordPress default for the post type posts. Maybe the renaming of the taxonomy to project-tags or something else would make sense. Otherwise the taxonomy will be mixed up by WordPress.

To show all tags you can use PHP:

the_tags('<ul class="taglist"><li>', '</li><li>', '</li></ul>');

Note: This function must be within the loop (archive.php or single.php).