There is a very wide variety of clustering methods, which are exploratory by nature, and I do not think that any of them, whether hierarchical or partition-based, relies on the kind of assumptions that one has to meet for analysing variance.
Having a look at the [MV] documentation in Stata to answer your question, I found this amusing quote at page 85:
Although some have said that there are as many cluster-analysis methods as there are people performing cluster analysis. This is a gross understatement! There exist infinitely more ways to perform a cluster analysis than people who perform them.
In that context, I doubt that there are any assumptions applying across clustering method. The rest of the text just sets out as a general rule that you need some form of "dissimilarity measure", which need not even be a metric distance, to create clusters.
There is one exception, though, which is when you are clustering observations as part of a post-estimation analysis. In Stata, the
vce command comes with the following warning, at page 86 of the same source:
If you are familiar with Stata’s large array of estimation commands, be careful to distinguish between cluster analysis (the cluster command) and the vce(cluster clustvar) option allowed with many estimation commands. Cluster analysis finds groups in data. The vce(cluster clustvar) option allowed with various estimation commands indicates that the observations are independent across the groups defined by the option but are not necessarily independent within those groups. A grouping variable produced by the cluster command will seldom satisfy the assumption behind the use of the vce(cluster clustvar) option.
Based on that, I would assume that independent observations are not required outside of that particular case. Intuitively, I would add that cluster analysis might even be used to the precise purpose of exploring the extent to which the observations are independent or not.
I'll finish by mentioning that, at page 356 of Statistics with Stata, Lawrence Hamilton mentions standardized variables as an "essential" aspect of cluster analysis, although he does not go into more depth on the issue.