Originally Drupal Core allows comment only nodes. Module User profile comments designed to extend this functionality for user profile.

User profile comments allows you display list of comments and comments form on the user page - user/[uid]. Also you can hide/display this content from user manage display in the Drupal Admin UI.

Drupal 8

Since in Drupal 8 comment settings have been converted from entity to the entity field, this module not needed in Drupal 8.
#731724: Convert comment settings into a field to make them work with CMI and non-node entities

Templates

You can use custom templates for comments:
comment--user-profile.tpl.php
comment-wrapper--user-profile.tpl.php

Supporting organizations: 
Development, Support

Project Information

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