What is the probability that on a specific day (e.g. December 24) two people have birthday in a room with 23 people?
The probability, given a specific day, one person has birthday is in my opinion $\frac{23}{365}$. So the probability for two birthdays is $(\frac{23}{365})^2$.
I believe that my calculation is not correct. If it is wrong, what is calculated by $(\frac{23}{365})^2$ in reality?
If you mean "exactly two people" the probability is $$\binom {23}2\left (\frac {1}{365}\right)^{2}\left (\frac {364}{365}\right)^{21}.$$
If you mean "at least two people" the probability is $$1-\left (\frac {364}{365}\right)^{23}-\binom {23}1\left (\frac {1}{365}\right)^{1}\left (\frac {364}{365}\right)^{22}.$$
Above no correction for leap years and deviations from the uniform distribution are taken into account.