I have seen people using"Many more happy returns of the day" to wish for birthday but my friends says "Many happy returns of the day" is common and natural.
Which one is common with more or without more?
The only time I can imagine seeing this would be on a Facebook page. If it were someone's birthday and lots of people had posted "Many happy returns of the day", it's just possible that people going on there later in the day might add "And here are many more happy returns of the day from me!"
However, in general everyday use, we do not use the word "more" in the phrase.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.