Data assimilation is a crucial task when monitoring the cardiovascular system. The goal is to exploit non-invasive measurements, such as medical images, in order to perform predictions. As all measurements, medical images provide a partial and noisy observation of the system state. When describing the mechanics of arteries or cardiac cavities, we rely on a system of parametric Partial Differential Equations (PDEs). In this work, we consider the case in which the domain in which the system is written (usually identified by using medical images) is uncertain. In view of accelerating Data Assimilation tasks, reduced-order models are often used. When the domain is moving or affected by uncertainties, the classical Model Order Reduction methods might have poor performances. In the present work, we are going to present two contributions: in the first one, a non-linear constructive non-linear manifold reduction method is presented in order to represent efficiently characteristic functions of varying domains; the second contribution exploits this method in order to build efficient fluid-structure interaction reduced-order models. Several theoretical results and numerical experiments will be presented