Each House has a unique/exclusive quest just before you reach the main story quest "Jackdaw's Rest." This explains why there's an achievement for each house upon finding the Map Chamber. The achievement itself should pop as you complete the main story quest "Jackdaw's Rest."
These four achievements for "Find the Map Chamber as a..." are going to involve creating four separate character saves. Completing the tutorial, selecting one of the four houses and progressing 2-3 hours through the main story questline. Until you meet Percival Rackham in the Map Chamber. You can skip every conversation and cutscene that makes up 80% of your standard playthrough by tapping or holding

.
Questline:
- The Path to Hogwarts
- Welcome to Hogwarts
- Charms Class
- Defence Against the Dark Arts Class
- Weasley After Class
- Professor Ronen's Assignment
- Welcome to Hogsmeade
- The Locket's Secret
- Professor Hecat's Assignment
- Secrets of the Restricted Section
- Tomes and Tribulations
- Herbology Class
- Potions Class
- The Girl from Uagadou
- Trials of Merlin
- Ollivander's Heirloom (Ravenclaw)
- Professor Hecat's Assignment 2
- Jackdaw's Rest
Credit Swearl:
You can skip 2 of the 4 tutorial runs by doing a
manual save as soon as you've been sorted into a house (e.g. Gryffindor) and arrive at their Common Room.
Then load the Autosave "The Path to Hogwarts" to complete the sorting hat segment again. This time picking a different house (e.g. Slytherin). Unfortunately after picking 2 houses the Autosaves no longer get you back to the sorting hat. You'll need to start a new save file to get the other two. Overall this will save you 20-40 minutes so no big loss if you didn't try this.