The following can be easily proved using perpendicular axes theorem and intermediate value theorem:

Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least one pair of mutually perpendicular lines that pass thru $P$ and lie on the same plane as $C$ such that the moment of inertia (MI) of $C$ about both lines are equal.

Symmetries in $C$ can lead to some points on $C$ being 'special' - such that w. r. to more than 2 lines thru them, MI of $C$ is the same; the center of a circular disk is obviously very special.

Question: Is it possible that for any planar convex $C$, there is at least one special point - a point with more than 2 lines passing through it giving the same MI for $C$? If such a special point always exists for any $C$, in the case when $C$ is a triangle, will it coincide with any known center of the triangle?

Remark: Above lemma won't hold for other moments - those defined with powers of distance from the axis different from quadratic. But analogous questions about concurrent lines yielding the same moment for $C$ could be asked - maybe with special points having only 2 lines of same moment passing thru them.

Note (January 2024): This question does overlap with this earlier question: On moments of inertia of planar and 3D convex bodies but the bit on the possible triangle center seems new.