Solution conflict: Expected number of distinct birthdays for $100$ people

The provided solution is correct. When it computes the chance that somebody has a birthday on Jan 1, it doesn't care how many people share the birthday. Then it says each day has the same chance of being somebody's birthday and uses the linearity of expectation.

We can see what is going on with smaller numbers. Say we throw two dice and ask what is the expected number of different numbers seen. We can do the problem directly by saying the first die is some number. The second die has $\frac 56$ chance of adding a new number, so the expected number of distinct numbers seen is $\frac {11}6$. This is less than $2$ because of the chance that the two numbers are the same. The approach in the solution you quote is to say the chance $1$ does not appear is $(\frac 56)^2$, so the chance it does appear is $1-(\frac 56)^2=\frac {11}{36}$. Then the expected number of numbers we see is $6 \cdot \frac {11}{36}=\frac {11}6$