Can someone help explain the behaviour of the implicit equation $x^n+y^n-n^x-n^y=0$, specifically the significance of the number 1.9667894071088...?

Let $$f_n(x,y)=x^n+y^n-n^x-n^y\qquad(n>0, \ x>0, \ y>0)\ .$$ The equation $$f_n(x,y)=0\tag{1}$$ defines a set $S_n$ in the first quadrant of the $(x,y)$-plane. Consider a point $(x_0,y_0)$ satisfying $(1)$. The implicit function theorem says the following: If $\nabla f_n(x_0,y_0)\ne(0,0)$ then there is a small window $W$ centered at $(x_0,y_0)$ such that $S_n\cap W$, i.e., the part of $S_n$ lying in $W$, is a smooth arc passing through $(x_0,y_0)$. When $\nabla f_n(x_0,y_0)=(0,0)$ then various things may happen, and the situation has to be studied in detail. E.g., we could have a selfintersection of $S_n$ at $(x_0,y_0)$.

We therefore compute $$\nabla f_n(x,y)=\bigl(n x^{n-1}-n^x\log n,n y^{n-1}-n^y\log n\bigl)=\bigl(g_n(x),g_n(y)\bigr)$$ with $$g_n(t):=nt^{n-1}-n^t\log n\ .$$ Plotting $t\mapsto g_n(t)$ for various $n$ shows that $g_n$ has two positive zeros $t_1$, $t_2$ (depending on $n$). This implies that the first component of $\nabla f_n(x,y)$ is zero along the two verticals $x=t_i$ and the second component is zero along the two horizontals $y=t_i$. We therefore obtain four points in the first quadrant where $\nabla f_n(x,y)=(0,0)$.

If it now happens that, by coincidence, the set $S_n$ contains one of these points then $S_n$ will probably be "special" there. In such a special situation the following three equations (in the variables $n$, $x$, $y$) will be fulfilled: $$f_n(x,y)=0,\quad g_n(x)=0,\quad g_n(y)=0\ .$$ You have found experimentally a triple $(n_0,x_0,y_0)=(1.9667892,3.2725,0.45562)$ where something special happens. Let's see at the numerics: