There are definately overlaps, but if you like the specifics of pumps and internal pipe design, mechanical is usually the best home for that focus. The real difference usually comes down to scale and purpose of the flow. Mechanical engineers focus on how a fluid interacts with a machine or a defined system to achieve a goal, like moving heat or generating power.
In my experiance, when I was designing a cooling system for a high-temperature fabrication line, my work involved calculating flow rates and pressure drops across heat exchangers—that's pure mechanical. We weren't focused on the chemical reactions happening, which is where the chemical folks step in. We were focused on the efficiency and the structural integrity of the pipes themselves.
Civil engineers are generally concerned with the large infrastructure aspect—think municipal water supply, storm drainage, or massive hydroelectric projects. They look at the fluid system as part of the surrounding environment and the built infrastructure. Mechanical is internal; civil is external. Chemical engineers, while they definitely do fluid dynamics, usually apply it within a complex process flow, often involving reactions. If your interest is in optimizing a massive industrial process where the fluid is also changing composition, that leans chemical. But if you want to design the actual components that move the fluid, like compressors, pumps, or ductwork for HVAC, mechanical is the route.
Choosing mechanical generally keeps the most doors open because you’re learning how physical systems operate end-to-end. It gives you a broad foundational understanding of physics and energy transfer that applies even if you eventually switch to manufacturing or controls.