= Solution
Write $S(\beta)=\sum_{ij}\mu_{ij}(\beta)$. Ignoring only terms independent of $\beta$, the Poisson <log-likelihood> is $\ell_P=\sum y_{ij}\log\mu_{ij}-S(\beta)$. The multinomial model is defined on the constraint $S(\beta)=n$, since its probabilities must sum to one. On this constraint its full <log-likelihood> satisfies
$$
\ell_M(\beta)=\ell_P(\beta)+n-n\log n+\log(n!),
$$
where both likelihoods include the same $-\sum\log(y_{ij}!)$ terms. Thus they differ by a constant on the admissible multinomial parameter set.
If the Poisson <maximum-likelihood estimator> has $S(\widehat\beta)=n$, it is feasible for the multinomial problem, and no feasible parameter can have a larger likelihood than it. Consequently it is also a multinomial maximum. With the usual uniqueness and identifiability assumption,
$$
\boxed{\widehat\beta=\beta^*,\qquad\widehat y_{ij}=y^*_{ij}.}
$$
Without uniqueness, the precise conclusion is that $\widehat\beta$ is a multinomial maximizer; arbitrary independently chosen maximizers need not be identical. This qualification is necessary for a general unspecified parametrization.
In a log-linear <generalized linear model> with an unrestricted intercept, differentiation with respect to that intercept gives $\sum(y_{ij}-\widehat\mu_{ij})=0$. Thus the fitted total automatically equals $n$. One can fit independent Poisson cell counts using the usual GLM machinery and recover the multinomial fitted probabilities as $\widehat\mu_{ij}/n$. The result is <constrained Poisson and multinomial likelihood equivalence>, not an assertion that the sampling experiments themselves are identical.
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