Birthday problem, want about using unordered counting?
When you have an event $E$ in a sample space $S$, the formula $P(E)=|E|/|S|$ only works when all the outcomes in $S$ are equally likely. When you forgot about order in the birthday problem, your sample space consisted of all multisets of $n$ birthdays. These are not all equally likely; the probability of two people being born on Jan 1 is $(1/365)^2$, but the probability of two people having the set of birthdays equal to $\{\text{Jan 1 , Jan 2}\}$ is $2(1/365)^2$, since there are two ways it can happen. This is why your formula fails.
The trick of forgetting the order often works in probability, but not here.
Side note: If you were to assert that all multisets of birthdays are equally likely, it would imply that the birthdays of the people are dependent in the following way. You can imagine the birthdays being chosen one at a time, where if there are already $n_i$ people whose birthday is the $i^{th}$ day of the year, the probability the next person is born on day $i$ is $$ \frac{n_i+1}{\sum_{i=1}^{365}(n_i+1)} $$ For example, the first person is equally likely to be born on any day, but then the second person has a $2/366$ chance of being born on the same day as the first person, and a $1/366$ chance of being born on any other day.