Help with wiring in an old house!
Retrofit ground is allowed, so you should be OK. There are 5 possible issues with that ground wire:

It is stranded (though rather thick strands). That is a bit unusual for ground. One of the pros can speak up as to whether that is a code issue or not.

It might be aluminum. For a bunch of reasons, that would be a problem. However, from the picture it appears that it may be either coated copper (in which case it is fine as long as you have a good connection) or just dirty (also fine, once you clean it off). If it is aluminum then the dissimilar metals problem comes up, but if that is the case then I think there are bigger problems as a single aluminum wire would not normally be installed as a retrofit ground, so that may be an indication of improper installation.

Where does it go? It must go back to something connected to the primary ground (i.e., in the main panel). That could be direct, or it could be via another circuit (which is OK). But you need to find out. While technically it may be OK, connection to a copper water pipe is not a great solution here, because plumbing could be changed at some time that would end up leaving the ground disconnected.

Color. Grounds are normally bare wire or green. There are exceptions at really large sizes, but that does not apply here. Electrically the color makes no differences (electrons are color-blind), but it does matter for code. From a practical matter, wrapping the exposed white area and the white area on the other end of the wire where it connects to the building ground takes care of making sure that the next owner doesn't get confused.

Is the ground working? Simplest way to check that is to check for continuity between the ground wire and the neutral (white in the cable) wire. But remember that even if you have continuity, the concerns noted above still apply.