Here is a simpler bound, based on the comment of R. van Dobben de Bruyn.

Let a solution of the equation be broken into two parts, c and d, where c is the n-smooth part of the solution, and is coprime to d, which of necessity is square free and has all prime factors bigger than n. (I leave the case n=1 to the reader.)  Then c is at most n! (2/1)(3/2)(5/4)...((n+1)/n)which for n greater than three is strictly less than (n+1)n!. So c is less than (n+1)! .

Turning to d, each prime divisor of d contributes at least one power of 2 when subject to Euler's phi, so d has fewer than n prime divisors. So d is less than e times n!.

Since the shrinkage under phi of the product is at most e(n+1), the original solution must be less than 3(n+1)!.  This is also a weak upper bound on the total number of solutions, but can probably be improved to show that the number  and location of solutions generally is less than (n+1)!, leaving the case of small n to the reader where all creation (counterexamples, arrghh spellcheck!) must lie.

**Edit 2020.07.09. GRP:**

The argument above for bounding $n$ given $m=\phi(n)$ is made even simpler, as $ n/\phi(n)$ is a product of 
$k$ many terms of the form $p/(p-1)$ where the $p$ are distinct primes.  This bounded above by $(2/1)(3/2 )(5/3)...$, which for all $k$ is less than $k+1$ and for large $k$ grows like $\log k$. Since $k$ is bounded by a function smaller than $\log m$, we can get an upper bound on $n$ that looks like $Cm\log\log m$, likely for $C$ less than 4.  Even when $k$ is large, $n$ can't have many more distinct primes than powers of 2 dividing $m$.

Towards the original question, note that there are easy solutions of totient value being a factorial, and that some of them can be extended by replacing certain powers of small primes by a prime $q$ such that $q$ is bigger than the base of the factorial and such that $q-1$ equals the powers of the small primes and $q$ is not already a prime factor of the solution being modified.
Thus it seems very likely that the number of solutions is not bounded as the size of the factorial grows.


**End Edit 2020.07.09. GRP.**

Gerhard "Leaving Hard Work To Others" Paseman, 2020.07.07.