= Solution
This is the <Poincaré residue> of a meromorphic top form with a simple pole. Write $z=z_1$ and locally $\alpha=(dz/z)\wedge\beta$, with $\beta$ a holomorphic $(n-1)$-form. The proposed residue is the pullback of $\beta$ to $Y$.
First it does not depend on the choice of $\beta$: if $dz\wedge(\beta-\beta')=0$, then at each point the difference has a factor $dz$, and its pullback to $Y$ is zero. Now change the local defining coordinate to $t=az$, where $a$ is a nowhere-zero <holomorphic function>. Such a factor exists because both coordinates define the same smooth <complex analytic hypersurface>. If $\alpha=(dt/t)\wedge\eta$, then
$$
dz\wedge\beta=z\alpha=\frac1a\,dt\wedge\eta
=dz\wedge\eta+\frac za\,da\wedge\eta.
$$
At points of $Y$, this is an equality in the ambient exterior-power fibre, and the last term is zero. It follows that $dz\wedge(\beta-\eta)=0$ there, so their pullbacks to $Y$ agree. This also covers changes of the tangential coordinates.
The restrictions are holomorphic top forms on $Y$, and their agreement on overlaps makes them a global section of the <canonical bundle>. Thus \b[the residue is independent of the adapted coordinates]:
$$
\boxed{\operatorname{Res}_Y(\alpha)\in\Gamma(Y,K_Y),\qquad
\operatorname{Res}_Y(\alpha)=\left.h\,dz_2\wedge\cdots\wedge dz_n\right|_Y.}
$$
The ambient-fibre comparison is important: pulling an ambient $n$-form directly back to the $(n-1)$-dimensional hypersurface would give zero and would not prove the required independence.
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