You don't "measure" your birthday, your birthday is a date and you measure your age by the passing of time. Yes, I did fill in "years old" because when you leave something out of a sentence, you've got to expect to be able to fill in the norm or you fail at communication. What's he suppoed to turn, 6 birthdays? 6 double rainbows?
It's a joke, but the joke is flawed in much the same way as saying "See you in a year" instead of "see you next year" at a new years eve party is flawed. I actually had several guys telling me "See you in a thousand years" december 31st 1999.