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The comparison of sets of objects is a research topic with applications in diverse fields such as computer science, biology and psychology. Since the introduction of the Jaccard index, many techniques have been proposed. This paper aims at extending an existing framework of comparison indices for sets. Firstly, the novel indices account for similarities between elements, rather than identity of elements as is the case for existing techniques. As a result, a richer framework of comparison indices is obtained. The use of fuzzy quantifiers in this framework is shown. Secondly, the machinery for sets is extended to the case of multisets, which results in two classes of comparison indices. The first class considers each element instance as a separate element, while the second class considers groups of elements instances as an atomic entity. The number of instances is then a property of this group, that is taken into account when calculating similarity between element groups. [ABSTRACT FROM AUTHOR]