But that isn't how the song is sung.

People always get mixed up regardless, people refer to people by different names like nicknames or relationship like 'mom' so when you get to the name part its always some funny mix

Yep, either way if you are singing Happy Birthday to her, you likely know her well enough to know her name well and sing it with no problem.

In Chinese, the first character (Tu) is the family name while the following character(s) are the given name (Youyou).

p.s. the "ou" sound in Chinese is pronounced more like an "o" rather than than an "ooh", so the joke doesn't really work (not quite, but it's close enough. I'm not very good at speaking Mandarin so take this with a slight grain of salt)

East Asian, China.

So her given name is "Youyou", pronounced Yoyo (flat high tone).

Youyou is her first name, and it's "dear", not "to". So it would be "Happy birthday, dear Youyou".