The translated and scaled copies of an equilateral triangle with fixed orientation are closed under intersection - the intersection is again an equilateral triangle with the same orientation.
What other convex shapes in 2D have this property?
The translated and scaled copies of an equilateral triangle with fixed orientation are closed under intersection - the intersection is again an equilateral triangle with the same orientation.
What other convex shapes in 2D have this property?
I think triangles (degenerated ones included) are the only such convex shapes. The idea is: If $C$ is such a convex shape, let $p_i\in\partial C$ for $i=1,2,3$ be three smooth points. One can obtain an approximate triangle as an intersection of three large copies $C$, say $m(C-p_i)+p_i$ for large $m$. Since these intersections are similar to $C$, and converge to a triangle as $m\to\infty$, $C$ itself is a triangle.
Since the shape $A$ is convex its boundary is differentiable almost everywhere. Take a point $p$ where it is differentiable. A line $l$ is tangent to A in $p$. Take two more points $q_1$ and $q_2$ with tangent lines $l_1$ and $l_2$, translate the shape by vectors $q-q_1$ and $q-q_2$ and intersect results. Suppose we get a shape $B$. The only point which can possibly have a tangent line parallel to $l$ will be $p$, but $\partial B$ is not differentiable there (unless all lines are parallel). Therefore, $B$ is not a homothetic image of $A$ and that means $B$ is at most one-dimensional.
$B$ being one-dimensional means $\angle (l_2, l) + \angle (l, l_1) \le \pi$ (we take angles which "look" at $A$). If there is at least 5 different directions for tangent lines we get a contradiction since in a convex pentagon the sum of angles is $3\pi$ but from our inequality it must be no more than $5\pi/2$. Therefore our shape is either a triangle (for which it is true) or parallelogram (false).
Because you do not provide a motivation for your question I do not know if the following reference is appropriate; however, you might want to look at George Stiny's book Shape: Talking about Seeing and Doing (Cambridge, Massachusetts: The MIT Press, 2006), in particular his discussion of closure on pp. 285-87 and p. 306.