There is an attractive theorem that says that if two plane figures
are *directly similar*, then so is any convex combination of them.
Below, $P_1$ and $P_2$ are directly similar polygons: they have
the same angles in the same sequence;
They are (positively) scaled, rotated, translated versions of one another.
The convex combination
illustrated $P_{12} = \frac{1}{3} P_1 + \frac{2}{3} P_2$ is
also directly similar.

(Let me henceforth abbreviate "direcly similar" with "similar.")
My question is:

Q1. Is there a natural extension to $\mathbb{R}^3$ and to higher dimensions?

Any extension cannot be an *exact* extension—some aspect has to give.
For example, here are two cubes $P_1$ and $P_2$, the latter
a $\frac{3}{4}$-scale rotated version of $P_1$.
Shown is $P_{12} = \frac{1}{2} P_1 + \frac{1}{2} P_2$,
and it is clearly not a cube.
I would define similarity in $\mathbb{R}^3$ to require all faces to
be similar with the same scale factor as well preserving all
dihedral angles—in other words, $B$ is similar to $A$ if
$B$ is a rotated, scaled, translated version of $A$.

But perhaps this is true?

Q2. Is the convex combination of two cubes (scaled, rotated, translated) always a parallelopiped? A parallelotope in $\mathbb{R}^d$?

Another approach would be to restrict the transformations:

Q3. Is there some condition on the transformations applied to the shapes that permits the similarity conclusion? Or: What is the widest class of transformations that leads to the similarity conclusion?

Certainly if $P_2$ is just a translated, scaled copy of $P_1$, then any convex combination is similar, in any dimension. Rotations are the culprit. But perhaps some rotations still lead to similarity.

It seems likely this has all been well-explored. If so, thanks for pointers!