The term search on the menu screen uses WP_Term_Query::get_terms(), which does not support slug searches. The terms list table search feature implements its own custom query that does support slug searches. Thus to accomplish what you want you either need to directly modify the SQL that get_terms() generates or override its query function all together and make your own custom query. The filters “terms_clauses” and “terms_pre_query” can be used to these ends.
One challenge in doing so is applying your modification only to the right query. get_current_screen() can be used to ensure you’re on the menu screen before adding your filter callback. Your callback can check the passed query object’s args (such as taxonomy) to further ensure you’re altering the the right query.
So obviously slug search is possible, though not so easy. Try searching for a plugin that’ll handle this for you. If none can be found, a custom coded solution will be necessary.
