How to list all tags except the current tag in tag.php

You may try get_tags with exclude parameter.
https://codex.wordpress.org/Function_Reference/get_tags

get_queried_object_id get the tag ID from tag page.

$tags = get_tags(array( 
    'exclude' => get_queried_object_id(), 
)); 

$tagList = array(); 

foreach($tags as $tag) 
{ 
    $tagList[] = '<a href="'.get_tag_link($tag->term_id).'">'.$tag->name.'</a>'; 

} 
echo implode(', ', $tagList);