menu_token module is working fine to achieve the purpose. Does the module
token_filter has this functionality? I can use tokens in wysiwyg using
token_filter but cannot make it working for menu title or url since there is no setting I could find. Since the comment in drupal issue says that it was solved using
token_filter module, I want to know the way it was achieved.
No, not by itself. The Token Filter module's functionality adds a token input filter to text formats in Drupal. The only field types that can use text formats are the 3 Text (formatted) field types, not the Link field type.
One can only speculate on the comment's solution you refer to. Text Formats/Input filters are designed to allow HTML input that should be filtered for presentation. It would be an odd-use-scenario to apply that to a link field.