I guess it would be similar to cleaning up an aluminum heat sink?
Never done that myself either but could be worth investigating

I'm guessing the modem is aluminum ? is it painted or coated with anything? Anodized ?

If it's anodized the only real way to fix it is to strip the Anodizing off and reanodize it witch is not all the hard to do but you would have to remove the labels you wanted to keep to do that .

If its nether painted or anodized I would just wet sand it with sand paper and sand in a straight line. there are a lot of you tube videos on how to sand aluminum for brushed finish.

This is a standard Hayes early modem case. There is no anodizing, or coloring. It appears to be just a brushed aluminum case. There might be something on it because it does not seem oxidized from age. Most of the case looks the same as the one I got back in the 80s to BBS. So I do not think it has a coating.
The white vinegar, and acetone (nail polish) had no effect on the surface or stain.

Anyone with some additional thoughts?

Bare aluminium oxidizes very quickly, but the film is very very thin, and stops.

This is a good thing, as the grey aluminium oxide layer halts further corrosion or chemical attack in most conditions.

If there is orange, black, green, or other staining on the metal, it's from metal cation exchange reactions, and the donated metal atoms then oxidized.

These will require the use of a strong abrasive (aluminium oxide is the same stuff 'emery paper' is coated with, and has the SAME mohs scale hardness!) to remove, because it *strongly* resists chemical attack, so normal oxide removing acids wont work.

Once the oxide layer is off, the metal itself is fairly soft, so brusing with a fine wire brush will return the finish.

Leave exposed to dry air for a day or so, and the protective oxide will re-form.

Many thanks for a reply.
The discoloring is white.
I added a couple pics to help.
It looks like it should wipe off when looking at the pictures. But this does not want to come off!
It has the same texture as the rest of the case. So there does not seem to be any etching.