I'm getting confused by the chi squared test of homogeneity, When performing the test, do the samples need to be distributed as multinomial?
The mention of multinomial made in your other question would be to the distribution within samples. The values across samples would be independent.
There are cases where the samples are dependent in both directions because both margins are being treated as fixed (in the 2x2 case this would be hypergeometric).
There are also cases that could be independent in both directions (Poisson counts); if you condition on the totals in one margin these would become multinomial.
For the chi-squared distribution to work you need the asymptotic multivariate normal to have the right form of dependence such that the chi-square statistic is equivalent to a quadratic form of the right kind to correspond to a sum of squares of independent standard normals of dimension equal to the df for the chi-square.