Your health insurance would be primary to you, but not because of the birthday rule.
The “birthday rule” applies to children and coordinates coverage for children who are listed on two parents’ group health insurance plans. It does not apply to spouses that are on each other’s health insurance job-based plans.
The birthday rule says that the health plan of the parent whose birthday is first in the calendar year is primary coverage for the children, and the plan of the parent with the later birthday is be secondary. The birthday rule goes by month and day, not year. So if you were born in June 5, 1975 and your wife was born in Jan. 8, 1977, her health insurance would be considered primary for your children.
But in your case, the health plan from your workplace would be primary to you, and your wife’s job-based health plan would primary to her. If you are listed on each other’s health plans, then that coverage would be secondary for each of you.
There are pros and cons to having two health insurance plans, so look over the costs and benefits before paying for double coverage.