How do I get the Mysterious Amulet, and what does it do?

Figured it out after much ado:

Mr. Miz shows with the quest "The Amulet" in the Lair of Infinite Agony. You don't need to do the raid boss first, but some have reported that the quest's likelihood of appearing on your minimap is random.

He gives you an option to purchase this ammy for about 125k. Equip it (functions as a Relic) for the "Hmmmmm" achievement. If you punch him, he will not let you buy it.

The amulet's stats are something like "Friendship +7" (the number varies). No one knows if this has any function.

If someone else in your party has completed the quest, you can backtrack (directions below) and purchase your own amulet from Miz directly.

_{Directions to Miz: From the entrance, take the elevator directly ahead. Make a hard right and open the door via the knob. In this room, turn left and head down a hallway with flying, flaming heads—Miz hangs out in this hallway. (YouTube video)}

_{Why I wasted several hours: I thought the quest wasn't spawning and tried resetting the game, fast-traveling between Santuary and Agony dozens of times, finishing the rest of the quests (not worth it, btw). Eventually I looked up the quest name ("The Amulet") in case someone in my party had inadvertently finished the quest during my first playthrough (before the raid boss). This was the case (quest was listed with completed quests). Even still, I was able to track Miz down in my own solo game (directions above) and buy the amulet from him.}

According to the Orcz Wiki's "Borderlands 2: Friendship" article:

[An] effect that is confirmed and recreatable is that, when worn in a party of 2-4 people in Coop, it will generate a pink cloud at the players' feet if they have the amulet equipped. With two people, pink and red heart particles appear, and yellow star-shaped particles will appear as soon as three or more people have it equipped. The cloud seems to be slowly growing in size and visual appearance over time, until it nearly fully covers the players' legs.

With the Commander Lilith & the Fight for Sanctuary DLC:

If the Mysterious Amulet is worn when feeding Butt Stallion in The Backburner, she will vomit up the Unicornsplosion, an effervescent shotgun. This is available once per playthrough.

^{Source: Borderlands wiki, "Butt Stallion" article
Also reported in this Reddit post by user, llaBocsiDcipE: Mysterious Amulet Use!}