'Aspect ratio' as a measure of the 'tall-and-skinny' property of unimodal distributions Kurtosis has not entirely lived up to being a measure of the 'tall-and-skinny' property. If you can access it, see Westfall 2014 for a valuable discussion of the pitfalls of using kurtosis as 'peakedness'.
Intuitively an aspect ratio of the height of a distribution divided by its width would convey a sort of 'tall-and-skinny' property. But since probability density functions (PDFs) have infinite width, such a value would be zero. It might be tempting to take the denominator to be the width at the average value of the PDF, but in general can lead to a factor of $\frac{1}{\infty - (- \infty)}$.
I remembered back to when I did chromatography that we instead used "width-at-half-height" to quantify the tall-and-skinny property of analyte peaks. Taking a look around online I found the notion of full width at half maximum (FWHM) that seems to be an equivalent term.

In correspondance with the diagram above, we could consider the ratio $\frac{f_{\max}}{|x_2 - x_1|}$ where $f$ is the probability density function for random variable $X$ with instances $x_1$ and $x_2$ corresponding to the width endpoints at half of the maximum of $f$.
At least, that is what I have thought on the matter so far. Any tradeoffs in choosing this as an estimate of 'tall-and-skinniness' for unimodal distributions?

You already got my +1 for drawing attention to Westfall (2014). A few thoughts:
Unimodality is of course important, and also quite a restriction. If you have multiple peaks, then all kinds of strange things can happen. And if you don't have a peak at all (e.g., a gamma distribution with shape $k<1$), then your $f_{\text{max}}$ is not even defined.
The next problem is when your density is noncontinuous. There may not be a point where it takes a value of $\frac{f_{\text{max}}}{2}$. But of course you can get around this by using suprema and infima as appropriate.
Here are a few well-behaved (that is, unimodal and symmetric) examples, all with your KPI value of $2$ (axes aligned for comparability):

I don't quite know whether I would call all of these "equally tall-and-skinny". Equally tall, yes, but not really equally skinny. Essentially, the two flanks rotate around their halfway point (indicated by red dots) between the four panels, and this constant halfway point is at the constant $x_{1,2}$. But of course all this is subjective.
These distributions all have the following form:
$$ f(x) = \begin{cases}
0, & x<-b \\
f_{\text{max}}\cdot\frac{b+x}{b-a}, & -b\leq x < -a \\
f_{\text{max}}, & -a\leq x<a \\
f_{\text{max}}\cdot\frac{b-x}{b-a}, & a\leq x <b \\
0, & b\leq x 
\end{cases} $$
for appropriate parameters $0\leq a<b$ and $f_{\text{max}}$. To ensure these integrate to $1$, we set for given $a$ and $b$
$$ f_{\text{max}}:= \frac{1}{a+b}. $$
To ensure that they have a prespecified value of your KPI
$$ \text{KPI} = \frac{f_{\text{max}}}{a+b}=\frac{1}{(a+b)^2} $$
(because $x_{1,2}=\pm\frac{a+b}{2}$), we use $a$ as a free parameter and set
$$ b := \frac{1}{\sqrt{\text{KPI}}} - a. $$
The plots have $a\in\big\{0,\frac{1}{10},\frac{2}{10},\frac{3}{10}\big\}$.
R code:
KPI <- 2
opar <- par(mfrow=c(2,2),las=1,mai=c(.5,.5,.1,.1))
    y_max <- sqrt(KPI)

    aa <- c(0,0.1,0.2,0.3)
    bb <- 1/sqrt(KPI)-aa
    f_max <- 1/(aa+bb)
    xx <- seq(-1.1/sqrt(KPI),1.1/sqrt(KPI),by=0.01)

    ff <- Vectorize(FUN=function(xx,aa,bb,f_max) {
        if ( xx < -bb ) return(0)
        if ( -bb <= xx & xx < -aa ) return(f_max*(bb+xx)/(bb-aa))
        if ( -aa <= xx & xx < aa ) return(f_max)
        if ( aa <= xx & xx < bb ) return(f_max*(bb-xx)/(bb-aa))
        if ( bb <= xx ) return(0)
        },vectorize.args="xx")

    for ( ii in seq_along(aa) ) {
        plot(xx,ff(xx,aa[ii],bb[ii],f_max[ii]),ylim=c(0,y_max),type="l",xlab="",ylab="")
        halfway <- c(-1,1)*(aa[ii]+bb[ii])/2
        points(halfway,ff(halfway,aa[ii],bb[ii],f_max[ii]),pch=19,cex=1.5,col="red")
    }
par(opar)

