Why is the dishwasher not recommended for my All-Clad MC2 line of pot and pans?

I can answer you with first hand experience and a picture. Your lovely dark gray finish will become light gray, streaked, and hideous. I'll never buy this style of pot again, it is just too useful to be able to dishwash them sometimes.

Like all aluminium they will discolour in the dishwasher, but will still be safe to use. One site I saw stated that the discolouration can be avoided by removing before the rinse cycle, which suggests to me it's the softeners rather than the detergent that does the damage.

It's worth noting that very acidic foods or prolonged cooking can also stain aluminium. Ours has a dark stain where the water came up to while steaming a Christmas pudding and I've seen some ruined by jam making.

NB in the past some people thought aluminium released into food during cooking was hazardous but this has since been disproven.

Dishwashing fluid isn't a simple soap; it's much more aggressive. A good portion of it is sodium hydroxide (lye) and sodium carbonate (washing soda) which will dissolve the protective anodized layer on aluminum surfaces (aluminum oxide), which is what holds the color.

Anodizing is a process that grows a layer of aluminum oxide on aluminum surfaces that can be dyed to form an attractive finish. The dissolution of the anodization is why Mike's pots look so bad now.

Sodium hydroxide and other bases can attack aluminum (metal), but it's not as apparent, especially when the concentration is fairly low. Remove a very thin layer of silver-colored aluminum and there's more silver-colored aluminum under it. Remove some of the anodization layer, and the dye can come out.