Yes, either wish is idiomatic and popular.
It means that I hope many good things will happen to you. So "Many happy returns on your birthday" means that I hope many good things happen to you on your birthday.
Here's what The Phrase Finder says:http://www.phrases.org.uk/meanings/244300.htmlMany happy returns
Have many more happy days, especially birthdays.Origin
Since the 18th century this has been used as a salutation to offer the hope that a happy day being marked would recur many more times. It is now primarily used on birthdays; prior to the mid 19th century it was used more generally, at any celebratory or festive event.
The first record O can find of it in print is in a letter written by Lady Newdigate in 1789 (published in Newdigate-Newdegate Cheverels
"Many happy returns of ye day to us my Dr Love."