The number of totient divisors of $n$ is $d(n-1)-d((n-1, \varphi(n-1))$. As $n$ gets large, then almost all $n$ have the property that $\varphi(n)$ is divisble by all small primes. The average number of prime divisors $p<y$ of $n-1$ is of magnitude $\log\log y$, hence, for almost all $n$ we have that the number of prime divisors of $(n-1, \varphi(n-1))$ tends to infinity. On the other hand the powerful part of $n-1$ is bounded, thus both $n-1$ and $(n-1, \varphi(n-1))$ are divisible by a large number of primes with exponent 1. Hence for almost all $n$ both $d(n-1)$ and $d((n-1, \varphi(n-1))$ are divisible by a growing power of 2, in particular, the number of totient divisors tends not to be prime.

The number of totient divisors of $n$ is $d(n-1)-d((n-1, \varphi(n))$. As $n$ gets large, then almost all $n$ have the property that $\varphi(n)$ is divisble by all small primes. The average number of prime divisors $p<y$ of $n-1$ is of magnitude $\log\log y$, hence, for almost all $n$ we have that the number of prime divisors of $(n-1, \varphi(n-1))$ tends to infinity. On the other hand the powerful part of $n-1$ is bounded, thus both $n-1$ and $(n-1, \varphi(n-1))$ are divisible by a large number of primes with exponent 1. Hence for almost all $n$ both $d(n-1)$ and $d((n-1, \varphi(n-1))$ are divisible by a growing power of 2, in particular, the number of totient divisors tends not to be prime.