Regardless of being made of copper or of plastic, a plumbing manifold functions as a control valve central to a plumbing system. It essentially connects the system in a fixed location that serves as a control box. Here hot and cold lines of water connect and are directed where they need to go, be it fittings or appliance. Essentially all water enters the manifold and is sent off where it needs to go at the manifold. This makes a plumbing system much more easily maintained, the pressure controlled throughout the system, and leaking lines easily shut off and repaired.
The deciding factor in the question, plastic or copper, is generally that of cost in proportion to quality.
Copper manifolds are far sturdier than plastic manifolds, they are less likely to crack under pressure, and they can stay undamaged when installed in a location that is directly exposed to light. Copper generally is more expensive, but can resist the pressure spike that results when, for example, a pipe freezes, much better than a plastic pipe. This makes the likelihood of replacement and repair much less. But, being a metal, copper is far more susceptible to corrosion, which can lead to cracking in the pipes, than plastic. As well, because most tubing is PEX, fittings will be required to connect, the two, as well as fittings for copper to copper connections. These fittings, being copper, are pricier than plastic. Furthermore, for copper manifolds, it is necessary to install gate valves to control the flow of water. But copper is a much stronger metal, and will last longer, if treated appropriately.
Plastic manifolds are much more versatile than copper, and often cheaper. They do not last as long, however. If left in sunlight, like rubber, plastic will crack. This may result in burst tubing, or leakage, which will require the manifold to be repaired. Also frozen plastic tubing is much weaker than copper, and so leaks and burst tubes in the winter may be an issue. But plastic is much easier to connect with plastic, and it is much simpler to connect an entire plumbing system of plastic to an entire plastic manifold. If indoors, it is likely the better investment to buy a plastic manifold, but only if it is taken care of and installed properly.