Attempt
Since this is a success/failure sample, I figured the event X : number of people who have birthday the chosen day follows a binomial distribution, with $ n = 1825$ and $p = 1/365$. Since $n \gg 1$ and $p \ll 1$, I can use an approach for the experiment via Poisson distribution with $λ=n \cdot p = 1825 \cdot (1/365) = 5$
$\Rightarrow X$~$P(5)$
$P[X>=1] = 1 - P[X < 1] = 1- P[X=0] = 1- e^{-5}$
Is this correct? I was told the answer should be 8.5%
Your calculation of $1-e^{-5}$ is the probability that a specific day has at least one person with that birthday. The question asks for the chance that every day has at least one person with that birthday. If we assume the days are independent the chance that every day has somebody with that birthday is $(1-e^{-5})^{365} \approx 8.478\%$. The assumption of independence is not correct because the fact that you know somebody has a birthday Jan 1 reduces slightly the chance that somebody has a birthday Jan 2, but it will be close.