Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - but not so sure of the approximation method used to generate $(x, y)$ pairs for $x^y = y$.
Is Desmos' approximation method known? Alternatively, what's an example method which'd yield the generated plot? Or is what's shown a valid "alternative solution" (despite it rendering $f(x)=x\uparrow \infty$ a non-function)?
Note red nears $x=1$ as $y \rightarrow \infty$.
There is nothing wrong with the $x^y=y$ plot by Desmos. The curve really does double back with respect to $x$ when $y$ goes above $e\approx 2.71828$. For instance, if we try $x=\sqrt2$ then we have both $(\sqrt2)^2=2$ and $(\sqrt2)^4=4$, so the curve should pass through $(\sqrt2,2)$ and then double back to catch $(\sqrt2,4)$. If you look near the right side of the left plot, just to the left of the nose, you see that the red curve actually hits both of these points.
If you meant to plot the limiting value of the power tower, it follows only part of the red curve, up to the nose at $y =e$. Beyond that the limit shoots up to infinity and the limiting power tower curve then looks like the blue one. It does not bend back because it can't; the reversed direction of the red curve above $y=e$ corresponds to the fixed point becoming unstable so the power tower cannot converge there. Thus the red and blue curves diverge not because the red curve is wrong (which isn't true) but because the limiting behavior stops following the red curve. The limiting value of $x$ is $\exp(1/e)\approx 1.445$.
Look to the left and you see the curves diverge again. This is because another form of instability, by oscillating instead of blowing up, sets in when $y$ drops below $1/e$. Again this causes the power tower to fail to converge; in this case the lower limit for convergence is $x=\exp(-e)\approx 0.066$.
This latter, oscillatory instability is not evident in any feature of the $x^y=y$ curve itself. But if we go to Desmos and plot the doubly iterated form -- $x^{x^y}=y$ -- we find there is an extra branch that looks like a badly drawn parabola when $x<\exp(-e)$. The power tower oscillates between the upper and lower parts of the "parabola", instead of trying to find the unstable fixed point of $x^y=y$ in this region. Thus whereas the "blow-up" instability for large $y$ is seen as the $x^y=y$ curve doubling back, the oscillating instability for small $x,y$ appears instead as that curve getting walled off.