Intutively, why does $x^{\frac{1}{\ln x}} = e$? For any $x \gt 0$, we have this odd identity:

$$x^{\frac{1}{\ln x}} = e$$.

You can see this by using the fact that $x = e^{\ln x}$.

I'm wondering if there's a good intuitive explanation for this one, given that $x^{\frac{1}{k}}$ is the operation that inverts raising $x$ to the $k$th power and $\ln x$ is the inverse of the exponential function. Is there some compelling intuitive or geometric argument that makes this identity more obvious than algebraic rearrangement?
The expression $\log x$ describes a number that is an exponent:  specifically, it is the exponent when, to which $e$ is raised, yields $x$.  When we speak of $1/\log x$, this is in some sense an inverse exponent:  for example, if I write $$a^b = c,$$ then $$c^{1/b} = a.$$  So $1/b$ is the exponent to which $c$ is raised to "recover" $a$.  Similarly, then, as $\log x$ is the exponent to which $e$ is raised to yield $x$, we must then have $1/\log x$ as the exponent to which $x$ must be raised to yield $e$.

Of course, all of the above is nothing more than a more prosaic restatement of the algebraic rules that you already wrote in your question.
$$x^{\displaystyle\left(\frac 1{\ln x}\right)}=e\iff \left(x^{\displaystyle\left(\frac 1{\ln x}\right)}\right)^{\ln x}=e^{\ln x}\iff x^1=x^{\ln e}=x^1\equiv{\bf T}$$