Only Google can answer that question. I would hope they make their apps, such as YouTube, only require authentication one time, then retain its login. That would be ridiculous to have to log in every time you use it. But they are the ones in control of the YouTube channel.
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Yes, if you already have 2FA it works fine. If you get auto-enrolled it will also work fine. It just pops up a verification on your phone which has the youtube app installed, and if you do not have this then it has fallback methods. If you do not have a phone then you're not going to be able to get much 2FA done and will have a much larger problem getting into the entire Google ecosystem.