In the US, it would be very puzzling. We would assume you mean the anniversary of your marriage, so why it would be your wife's anniversary but not yours would be a mystery.
There are other uses - for example, at my company, you get a gift on various milestone employment anniversaries, so if at a meeting, someone said "Today we want to recognize Kim's 5-year anniversary" we would know it meant she joined the company 5 years ago.
But it's never used for a birthday.
Student or Learner