Does the following proposition hold in number theory

No, this is not true.

Take $n=18$. The (proper, not including $1$ or $18$) divisors are $\{2,3,6,9\}$. The only such divisor which $2$ divides or which divides $2$ (besides itself) is $6$, while the only such divisor for $9$ is $3$. So, the sum is $\phi(6)=\phi(3)=2$ for either $2$ or $9$. In general, for an odd prime $p$, the divisors $2$ and $p^2$ of $2p^2$ exhibit the same property.

There are more "exotic" examples as well, including the first odd counterexample, $297$. Here, the divisors $9$ and $11$ lead to equal sums. The general pattern here is that $297=p^3q$ with $p^2+2=q$, and it is not known that there are infinitely many pairs of primes with this pattern.