Elementwise maximum of two positive definite matrices
No. A conterexample: take a small $\epsilon>0$ and define $$ A=\begin{bmatrix}1 & 1-\epsilon & 0\\1-\epsilon & 1 & 0\\0 & 0 & 1\end{bmatrix},\qquad B=\begin{bmatrix}1 & 0 & 0\\0 & 1 & 1-\epsilon\\0 & 1-\epsilon & 1\end{bmatrix}. $$ Both matrices are positive definite, but the element wise maximum $$ C=\begin{bmatrix}1 & 1-\epsilon & 0\\1-\epsilon & 1 & 1-\epsilon\\0 & 1-\epsilon & 1\end{bmatrix} $$ has a negative determinant (for small $\epsilon>0$).
I typed almost the same answer A.Γ. so here is a concrete example instead
A
array([[22., 9., 7., 6.],
[ 9., 14., 5., 8.],
[ 7., 5., 12., 11.],
[ 6., 8., 11., 14.]])
B
array([[20., 2., 12., 2.],
[ 2., 20., 2., 5.],
[12., 2., 12., 4.],
[ 2., 5., 4., 12.]])
C
array([[22., 9., 12., 6.],
[ 9., 20., 5., 8.],
[12., 5., 12., 11.],
[ 6., 8., 11., 14.]])
with eigenvalues
eigA = [ 1.30323419, 7.88111427, 13.88897809, 38.92667345]
eigB = [ 2.81884072, 9.73820335, 20.42843182, 31.01452411]
eigC = [-0.11100762, 11.43648115, 13.68814039, 42.98638609]