Though it may seem odd, it is possible the plumber knows what he's talking about. If this is a Polybutylene pipe, it could indeed be brittle and fail if it's jostled during the installation of the filter.
It's impossible to tell from the picture what type of pipe that is, but based on the plumbers concern I'd guess it is Polybutylene.
Polybutylene
Polybutylene was used in pipes between about 1980-1994/5, and was commonly used as supply line in the 80's. It was found (in some cases) to react with oxidants in the water supply, which caused the pipe to become brittle and fail. To my knowledge, this type of piping is no longer used. Though if the house was built; or there was supply line work done, during the 80's, it's likely this is Polybutylene pipe.
High Density Polyethylene
Don't confuse Polybutylene pipe with Polyethylene pipe. High-density polyethylene is commonly used as water supply line today, and is stable and safe to use.
Cross-linked polyethylene
Cross-linked polyethylene (PEX) is not typically used as an underground supply line, as it can sometimes react with minerals in the soil. Where PEX is used underground, it's typically encased in sand (or other non-reactive materials) to prevent negative reactions. Because of this it's often more expensive to bury PEX, so other materials are used instead.
How can I tell what it is?
The only way to be sure what type of pipe this is, is to look for markings on the pipe itself. Polybutylene will be marked PB####
, Polyethylene will be marked PE####
, Cross-linked polyethylene will be marked PEX-x
(where x = A, B, or C).
The next step secret to getting more of this faucet apart are the two slots cut across the cupped part. See the arrows in the following picture.
It looks to me as that the cupped part is threaded down inside the larger diameter part that sticks up from the hole in the sink. There is probably a special tool used to engage these slots to thread the part out. It may be possible to lay a piece of metal bar stock across these slots as a tool. If you do that make sure that it has nice square corners.
Best Answer
In the strictest sense, if it has a metallic finish, it is metal, though it may only be a few molecules thick over a plastic substrate. But you refer to the base material. It can be hard to tell when it's in place due to the resistance of the mechanism masking the apparent mass. The way to distinguish without destructive sampling is essentially it's mass. Metal is denser than plastic. Removing the handle will usually make it easy to to tell just by weight. Many modern faucets are often made of both metal and plastic parts though.
One can still be fooled by weight. A solid, quality plastic handle may weigh more than a hollow, thin, cheap brass handle. The only sure way is to take a sample by drilling a small hole in an inconspicuous place. The shavings could be tested in a lab, after which they could determine exactly the type of resin or metal composition, if you paid them enough.
It usually suffices to simply hold a flame to the shavings. Plastic burns, metal does not.