Many Japanese textbooks have this kind of expressions. Some say they are outdated, some says they are used only in formal occasions. If you hear that expression, please let me know in what cases you hear? Are there other alternate greetings or phrases to start a conversation with?
Now, I think this is used more when parting the company of someone you just met. “It was nice meeting you.” or “Nice meeting you.” Or, in a follow up email or text, “Hi John. It was nice meeting you yesterday.”
These are part of polite introductions and can still be used as part of a first greeting after hearing someone’s name and shaking their hand. Often in more casual introductions especially when there are a number of people being introduced, you would just repeat your own name, or say for example “Hi John”. When leaving a gathering saying goodbyes to someone you have just met you often say “It was nice meeting you”, replied with “Nice to meet you too”. I have never heard “You too” as part of these exchanges.
I agree with the above two comments. When saying goodbye to someone you have just met, it’s not unusual to say “It was nice to meet you,” and for them to reply, “It was nice to meet you, too!” The extra “You, too” is not necessary.
Here’s a real-life exchange in English on a Russian-language blog. The gentleman is elderly, but younger people raised to be polite will be the same.
(At 15:28 there is no dialog recorded, but none needed: https://youtu.be/-R_4wnIlSEI?t=15m28s )