In the literature there are two different approaches that extend the classical crisp notion of equivalence relation to the fuzzy framework. On the one hand, one can find the notion of indistinguishability operator and a few of its generalizations. These can be understood as a kind of measurement of the degree of similarity or indistinguishability between objects. On the other hand, fuzzy (quasi-)metrics measure such a degree with respect to a parameter. The study of both types of the aforesaid notions has been carried out independently without any connection between them. As a consequence, the notion of modular indistinguishability operator has been introduced recently. Such a notion unifies under the same framework both aforesaid similarity concepts. In this paper, we explore the aggregation problem for modular indistinguishability operators and for several generalizations. Hence we introduce the notions of modular fuzzy pre-order, modular fuzzy partial order and modular equality and we characterize the functions that are able to fuse all these different types of modular similarities. The aforementioned characterizations are stated in terms of triangular triplets or related notions, monotony and dominance. In contrast to the non-modular case, the class of those functions that merge modular fuzzy pre-orders (modular fuzzy partial orders) is shown to match the class of modular indistinguishability operators (modular equalities). Furthermore, the relationships between the non-modular aggregation problem, the modular one and the fuzzy metric aggregation problem are explored and the differences between them are clarified by means of appropriate examples.