You can follow https://www.shamimsplugins.com/support/topic/preview-or-clickable-profile-avatar-so-you-know-who-is-messaging-you/#post-34361

Let me know.

add following code in your theme’s (child theme’s if you are using) functions.php

add_filter( 'fep_filter_user_name', function( $name, $id ){
	if ( $name && fep_get_the_id() ) {
		$name = '<a href="https://yoursite.com/'. fep_get_userdata( $id, 'user_nicename', 'id' ) . '/">' . $name . '</a>';
	}
	return $name;
}, 10, 2);

So you want to show link to profile only in this attachment marked names, right?

add following code in your theme’s (child theme’s if you are using) functions.php

add_action( 'fep_message_table_column_content_author', function(){
    if( 'threaded' === fep_get_message_view() ){
		?><span class="fep-message-author"><a href="https://yoursite.com/<?php echo fep_get_userdata( fep_get_message_field( 'mgs_last_reply_by' ), 'user_nicename', 'id' );?>/"><?php echo fep_user_name( fep_get_message_field( 'mgs_last_reply_by' ) ); ?></a></span><span class="fep-message-date"><?php echo fep_get_the_date( 'mgs_last_reply_time' ); ?></span><?php
	} else {
		?><span class="fep-message-author"><a href="https://yoursite.com/<?php echo fep_get_userdata( fep_get_message_field( 'mgs_author' ), 'user_nicename', 'id' );?>/"><?php echo fep_user_name( fep_get_message_field( 'mgs_author' ) ); ?></a></span><span class="fep-message-date"><?php echo fep_get_the_date( 'created' ); ?></span><?php
	}
});

Did you removed previous code then added latest code? You should remove previous code.
Let me know.