This achievement is for collecting 20
different Legendary Outfits. E.g., if you have already collected Eulogy Jones's suit and you later get Eulogy Jones (wearing his namesake suit), the achievement does not credit the second instance of Eulogy Jones's suit. This also means that you cannot craft the same Legendary Outfit repeatedly in the Outfit Workshop to get the achievement.
Several people are noting that they are getting progress from multiple appearances of the same legendary outfit, so this may be incorrect. I was basing this on my personal experience noted above about getting Eulogy Jones's suit and Eulogy Jones through separate lunchbox cards did not show the additional progress on the Xbox app, but I may have been mistaken.
You can get legendary outfits in four ways:
1) Finding legendary loot when exploring or on quests. Some quests will include Legendary Outfits and Legendary Weapons as rewards for the quest. Usually, when a legendary item is included in a quest, it will be shown in the quest screen, but I have also recovered legendary weapons and outfits on the intermittent exploration quests. When a Legendary Outfit is included on a Quest, it is either indicated by showing the actual outfit as a reward (e.g. Jobinson's Jersey) or is shown as a gold outfit icon to indicate that some Legendary Outfit will be rewarded.
2) Crafting outfits in the Outfit Workshop. Once you have enough dwellers, you can upgrade the Outfit Workshop to build Legendary Outfits. You will need to find recipes for the outfits on quests or on exploration quests. Legendary Outfits each require several pieces of legendary junk to build, as well as caps and time.
3) Legendary Dwellers come wearing a Legendary Outfit. All Legendary Dwellers come wearing a Legendary Outfit,
but the outfit will only count toward this achievement if you have not previously collected that outfit. Unclear if this part is true.
4) Lunch boxes contain Legendary Outfits as rare cards.
The same rules apply to duplicates. Uncertain if this is true.