This is related to the Doomsday rule. Certain days of the year, such as 4/4 and 6/6 are always on the same day of the week as each other in a given year.
A first approximation would be $\frac17$.
But if one person was born in January or February, and the second person was born after February, then there is no way that they would always have their birthdays on the same day of the week on both leap years, and non-leap years.
If they are both in the January to February range, or they are both in the March to December range, then it's OK.
This restriction brings the odds down a little lower than $\frac17$ but how much lower?
I'm not sure what to do with people who were born on leap day. So I guess we can leave them out. Solve for two people where neither was born on leap day.
Best Answer
If it is impossible to be born on 29 February and other dates are equally likely then in January and February there are $59=8\times 4+9 \times 3$ days and in the other ten months there are $306=43\times 2+44 \times 5$ days
so the probability would be $\dfrac{8^2\times4+9^2\times3+43^2\times2+44^2\times5}{365^2} \approx 0.10416$, substantially less than $\frac17 \approx 0.14286$
If it is possible to be born on 29 February (say with $\frac14$ the probability of other dates) and other dates are equally likely then perhaps the probability would be $\frac{8^2\times4+9^2\times3+0.25^2\times1 +43^2\times2+44^2\times5}{365.25^2}\approx 0.10402$, which is barely changed