Does ONLY the ellipse have these properties? Two parallel lines are tangent to an ellipse. Between those two lines, every line parallel to those two intersects the ellipse in two points.
The precise midpoint between those two points lies exactly on the line connecting the two points of tangency.
My question here is whether that last proposition is true for every pair of parallel tangents only for the ellipse and no other shape.
PS: For present purposes, let's define "tangent" as "touching but not crossing the boundary." Then an oblique line through the corner of a rectangle is a "tangent", and we see that the rectangle is NOT an example of another shape with the property of interest.
PPS: ok, Let's refine the statement of the problem a bit. Consider a closed bounded set with non-empty interior in the plane. Suppose it is strictly convex, i.e. every point between two of its points is one of its interior points. This entails that a line that intersecting its boundary but not its interior intersects it at only one point. Call such a line a tangent line. It follows that for every tangent line, there is exactly one other tangent line parallel to it. Suppose that for every line parallel to those two and between them, the midpoint of the intersection of that line with our closed bounded convex set is on the line connecting the two points of tangency.
Does it follow that our closed bounded set is the convex hull of an ellipse?
 A: Every closed convex piecewise-differentiable curve with the given tangent property is an ellipse.
Proof: The problem is affine, in the sense that if a curve has the given property then so does any affine transformation of it. So, starting with a pair of tangents at the widest extent of the curve, use a rotation to make the tangents vertical and a shear to bring the curve to  $\mathcal{C}$ whose line of symmetry is the $x$-axis.
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Now take the horizontal pair of tangents on  $\mathcal{C}$, meeting it at two points one vertically above the other. Translate it so this vertical line is the $y$ axis. Then   $\mathcal{C}$ is symmetric about both the $x$ and $y$ axes. Scaling along these axes brings their intercepts to $1$. Every other point has radius at most $1$, by the way the original tangents were chosen.
Proposition 1.  $\mathcal{C}$ is balanced, i.e., $x\in \mathcal{C}\implies -x\in\mathcal{C}$.
This follows directly from the symmetry along the two perpendicular axes.
Hence given any pair of tangents, the line joining the points of contact passes through the origin.
Proposition 2. The curve is differentiable.
Join opposite corners by a line through the origin. Then $\mathcal{C}$ would have equal distances from this line along two sets of parallel lines, which gives a contradiction.
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Proposition 3. Any point on $\mathcal{C}$ with radius $1$ has a perpendicular tangent.
A point with the maximum radius $r(\theta)=1$ must have $r'=0$.
Proposition 4. If $OA$ and $OB$ have radii of $1$ then so does their angle bisector $OC$.
The tangent parallel to $AB$ touches the curve at some point $C$. The line $OC$ cuts $AB$ in half by hypothesis and is thus the median and angle bisector of $AOB$, and perpendicular to $AB$. Thus $\mathcal{C}$ is symmetric about $OC$ and so the tangent at $C$ is perpendicular to $OC$.
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Let the tangent at $C$ meet the tangent at $A$ at the point $P$. Consider tangents parallel to $AC$ and the line $Q'OQ$ joining the opposite tangents. This line passes through the midpoint of $AC$ by hypothesis. In the limit, nearby points $A'$ on $AP$ and $C'$ on $CP$ with $A'C'$ parallel to $AC$ are also bisected by $OQ$ since $AP$ and $CP$ are tangents to $\mathcal{C}$. But this means that $OQ$ is the median of $APC$, and thus $Q$ is on $OP$. Since $OAPC$ is a cyclic quadrilateral with diameter $OP$, the bisected chord $AC$ is perpendicular to $OP$ and so $OC=OA=1$.
Proposition 5. $\mathcal{C}$ is a circle.
Since the $x$ and $y$ intercepts have radius $1$, one can keep taking the angle bisectors, forming a dense set of points of radius $1$. By continuity, all points have the same radius.
Hence the original curve is an affine transformation of a circle, namely an ellipse.
