Welcome to Android Central! Like all other Google apps on an Android phone, the very fact that you're logged into the phone with your Google account means you're automatically logged into your Google apps (including YouTube). You can add another Google account and switch between those accounts, or go into Incognito Mode, but I don't believe there's a way to sign out of the YouTube app (just like you can't sign out of the Gmail app, for example).
I suppose you could try going to Settings>Apps, selecting YouTube, then Storage, then Clear Cache/Data, but I think when you open it again, it will automatically sign you back in.