The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are given by $\frac{n!}{(2!)^{e_2}(3!)^{e_3}\ldots (n!)^{e_n}}$, e.g.,
$$M_4(u_1,u_2,u_3,u_4) = u_4 + 4 u_1 u_3+ 6 u_2^2 + 12 u_1^2 u_2 + 24u_1^4 .$$
The monomials encode partitions of $n$, i.e., $n = 1e_1 +2e_2 + \cdots + ne_n$ for each monomial of $M_n(u_1,...,u_n)$.
These coefficients appear on p. 311 of Counting Surfaces by Eynard as enumerating the intersection numbers associated with cotangent bundles of $\overline{M}_{0,m}$, the compact moduli space of stable curves of genus $0$, with $m$ marked points (see pgs. 252-3 of Eynard). They also occur as coefficients of the Jack symmetric functions $J_[n]^\alpha$ and the expansions of $(x_1 + x_2 + \cdots +x_{n+1})^n$ in the monomial symmetric polynomials (see A036038 and the MOPS link pg. 4, therein).
The coefficients of the same monomials of the ParPs $[N]$ enumerating noncrossing partitions and several other geometric constructs (see OEIS A134264 and this MO-Q) are given by $\frac{n!}{[n+1-(e_1+e_2+\cdots +e_n)]! (e_1)! (e_2)! ... (e_n)!}$, e.g.,
$$N_4(u_1,u_2,u_3,u_4) = u_4 + 4 u_1 u_3+ 2 u_2^2 + 6 u_1^2 u_2 + u_1^4 $$ .
The Hadamard product
$$(1,4,2,6,1)*(1,4,6,12,24) = (1,16,12,72,24)$$
generates the coefficients of the ParPs of A248927 (a refinement of A141618), e.g.,
$$P_4(u_1,...,u_4) = u_4 + 16 u_1 u_3+ 12 u_2^2 + 72 u_1^2 u_2 + 24 u_1^4.$$
$[N]$ gives the coefficients of the o.g.f. that is the compositional inversion of the formal o.g.f. $O(x) = x + c_1x^2 +c_2x^3 + \cdots$ in terms of its shifted reciprocal $x/O(x) = 1 + u_1x + u_2x^2+\cdots$ while $[P]$ gives an analogous result but for the coefficients of the the e.g.f. $g(x)/x$ where $g(x)$ is the compositional inverse of $f(x)$ in terms of the coefficients of the e.g.f. of its shifted reciprocal $x/f(x) = 1 + u_1x + u_2 \frac{x^2}{2!} + u_3\frac{x^3}{3!} + \cdots$.
Is there an overarching geometric interpretation of (or deeper algebraic intuition underlying) the Hadamard product identity?
In the title of the post, I use noncrossing partitions, but only as an indication of the flavor of the question and not as a restriction on the numerous potential geometric constructs that might be used in an answer.
