The intricate morphology of brain vessels poses significant challenges for automatic segmentation models, which usually focus on a single imaging modality. However, accurately treating brain-related conditions requires a comprehensive understanding of the cerebrovascular tree, regardless of the specific acquisition procedure. Our framework effectively segments brain arteries and veins in various datasets through image-to-image translation while avoiding domain-specific model design and data harmonization between the source and the target domain. This is accomplished by employing disentanglement techniques to independently manipulate different image properties, allowing them to move from one domain to another in a label-preserving manner. Specifically, we focus on manipulating vessel appearances during adaptation while preserving spatial information, such as shapes and locations, which are crucial for correct segmentation. Our evaluation effectively bridges large and varied domain gaps across medical centers, image modalities, and vessel types. Additionally, we conduct ablation studies on the optimal number of required annotations and other architectural choices. The results highlight our framework's robustness and versatility, demonstrating the potential of domain adaptation methodologies to perform cerebrovascular image segmentation in multiple scenarios accurately. Our code is available at https://github.com/i-vesseg/MultiVesSeg.