If you have android mobile and use youtube app. If you want no save youtube history. Then Update youtube app use incognito mode. Incognito Mode is already launch in Chrome Browser. Now available in youtube app. you can also use in mobile chrome app.
What is Incognito Mode?
If you use Incognito mode then no save your History. Private browsing, privacy mode or incognito mode is a privacy feature in some web browsers to disable browsing history and the web cache