= Solution
An <irreducible complex analytic hypersurface> is a closed analytic subset of pure complex codimension one which cannot be expressed as the union of two proper closed analytic subsets. Here a <local defining function of a complex analytic hypersurface> means a reduced local equation: a <holomorphic> germ generating its vanishing ideal, with each local branch occurring once. The word reduced matters. Merely prescribing the zero set would allow both $f$ and $f^2$, and the requested uniqueness would then be false.
The local facts used are these: in <holomorphic coordinates> the <local ring> $\mathcal O_{X,p}$ is $\mathbb C\{z_1,\ldots,z_n\}$, a <regular local ring> and a <unique factorization domain>; its units are exactly the germs nonzero at $p$; and the analytic Nullstellensatz identifies the zero-set ideal of a germ $f$ with $\sqrt{(f)}$. A reduced germ is a product of distinct irreducible factors. <Unique factorization> makes its <principal ideal> radical: if $f\mid h^r$, every irreducible factor of $f$ divides $h$, so $f\mid h$. Thus two reduced germs with the same hypersurface germ have $(f)=(g)$. Write $g=uf$, $f=vg$. The <local ring> is a domain, so $uv=1$. Hence $u$ is a <holomorphic> unit and $u(p)\ne0$. This proves <reduced local defining functions differ by a holomorphic unit>. Global irreducibility does not imply local irreducibility at a singular point; the same argument uses the product of all local branches and therefore covers that case too.
A <divisor on a complex manifold> is a locally finite integer combination of irreducible analytic hypersurfaces. On a <compact> manifold only finitely many have nonzero coefficients. Choose local meromorphic equations $f_i$ for a divisor $D$, by products of powers of reduced local equations. Their ratios are <holomorphic> units on overlaps. For the <holomorphic line bundle associated to a divisor>, choose frames satisfying
$$
e_i=\frac{f_j}{f_i}e_j.
$$
The ratios obey the cocycle identity and so define a <holomorphic line bundle> $[D]$. This convention makes the local meromorphic sections $f_i e_i$ agree, with divisor $D$. Changing the equations by units changes frames and gives an isomorphic bundle.
For hyperplanes $H_1,H_2$ in <Complex projective space>, choose nonzero linear equations $\ell_1,\ell_2$. The degree-zero quotient $r=\ell_1/\ell_2$ is a global <meromorphic function> and has divisor $H_1-H_2$. A <principal divisor on a complex manifold> gives a trivial associated bundle: using the same global meromorphic equation on every chart makes all frame transitions equal to one. The allowed tensor-product rule now gives
$$
[H_1]\otimes[H_2]^{-1}\cong[H_1-H_2]\cong\mathcal O,
\qquad\boxed{[H_1]\cong[H_2].}
$$
If the hyperplanes coincide the same conclusion is immediate. Equivalently, their common bundle is the hyperplane bundle $\mathcal O(1)$.
Locally at the centre of a <blowup of a complex manifold at a point>, replace a coordinate neighbourhood in $\mathbb C^n$ by
$$
\widetilde U=\{(z,[\ell])\in U\times\mathbb{CP}^{n-1}:z_i\ell_j=z_j\ell_i\text{ for all }i,j\},
\qquad\sigma(z,[\ell])=z.
$$
Away from zero the direction is uniquely $[z]$, so this map is an isomorphism there; glue it to $X\setminus\{p\}$. In the chart $\ell_j\ne0$, write $z_j=u$ and $z_i=ut_i$ for $i\ne j$. These are smooth <holomorphic coordinates> on the blowup. The <exceptional divisor> is $u=0$, globally $E=\sigma^{-1}(p)\cong\mathbb P(T_pX)$.
The <canonical bundle formula for a point blowup> is
$$
\boxed{K_{\widetilde X}\cong\sigma^*K_X\otimes[E]^{\otimes(n-1)}.}
$$
The local chart also explains the exponent: pulling back a coordinate volume form gives $\pm u^{n-1}du\wedge\bigwedge_{i\ne j}dt_i$, so the Jacobian section has divisor $(n-1)E$.
Since $Y$ is smooth at the centre, choose coordinates with $Y=\{z_1=0\}$. Its <strict transform> is a genuine analytic hypersurface. In a blowup chart $j\ne1$, it is the closed set $t_1=0$: off $E$, the original equation $ut_1=0$ reduces to $t_1=0$, and taking the closure restores the points with $u=0$. In chart $j=1$, there are no such off-exceptional points, and the <strict transform> is empty there. These descriptions agree on overlaps and show analyticity, with $\widetilde Y\cap E\cong\mathbb{CP}^{n-2}$. The total transform has the extra factor $u$ exactly once, so
$$
\sigma^*Y=\widetilde Y+E,
\qquad[-\widetilde Y]\cong\sigma^*[-Y]\otimes[E].
$$
Under the given anticanonical assumption, this yields
$$
\boxed{[-\widetilde Y]\cong K_{\widetilde X}\otimes[E]^{\otimes(2-n)}.}
$$
To prove the only-if direction, one must rule out an accidental trivial power of $[E]$. The local blowup is the tautological <line bundle> over $\mathbb{CP}^{n-1}$, with $E$ its zero section. Its <normal bundle> is therefore $\mathcal O_E(-1)$. The <normal bundle of a smooth analytic hypersurface> identifies $[E]|_E$ with this bundle. If $n\ge2$, restrict to a <projective line> inside $E$; the restriction of $[E]^{\otimes(2-n)}$ has degree $n-2$. A trivial <line bundle> has degree zero, so it is trivial only if $n=2$. Conversely the displayed tensor factor is literally trivial when $n=2$. This is the <anticanonical strict transform under a point blowup> criterion:
$$
\boxed{[-\widetilde Y]\cong K_{\widetilde X}\ \Longleftrightarrow\ n=2.}
$$
For completeness, dimension one cannot satisfy the premise in this <compact> setting: an irreducible analytic hypersurface is one point, and $[-Y]$ has degree $-1$ on its curve component, whereas $K_X$ has degree $2g-2$. Thus no exceptional dimension-one case invalidates the equivalence.
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