Metal turned dull grey in dishwasher?
It's cast aluminum.

Most dishwasher detergents contain lye (NaOH, sodium hydroxide) - which reacts drastically with aluminum. Fortunately the dwell time in a dishwasher is short enough that it generally only darkens, as opposed to dissolving entirely (which it will do with adequate time.) 

Yes, it's safe to use (after hand-washing and rinsing throughly), no, it's not dishwasher-safe and should be hand-washed instead.

The filter cup is of some different metal (either stainless steel or something nickle-plated, at a guess.)