They ought to have the same type.
I think she(Esther)'s an sp3w4. This is at least a 3 fix:
"I felt like a racehorse in a world without racetracks or a champion college footballer suddenly confronted by Wall Street and a business suit, his days of glory shrunk to a little gold cup on his mantel with a date engraved on it like the date on a tombstone."
What triggered her mental breakdown seemed to me (the book doesn't tell) to be her enormous expectations combined with a feeling of inadequacy triggered by not being accepted into the prestigious writing course and lacking skills like German. "The trouble was, I had been inadequate all along, I simply hadn't thought about it." The first reaction is making a new plan for the summer, writing a novel, learning new skills, anything not to waste her time, but she can't because of the depression, she can't write anymore and she loses her confidence.
Esther seems to merge with multiple characters whose approval she wants, mainly Doreen and Jay Cee.
"The only thing Doreen ever bawled me out about was bothering to get my assignments in by a deadline. “What are you sweating over that for?” (...) “Jay Cee’s ugly as sin,” Doreen went on coolly. (...) Jay Cee was my boss, and I liked her a lot, in spite of what Doreen said. She wasn’t one of the fashion magazine gushers with fake eyelashes and giddy jewellery. Jay Cee had brains, so her plug-ugly looks didn’t seem to matter. She read a couple of languages and knew all the quality writers in the business. (...) Jay Cee wanted to teach me something, all the old ladies I ever knew wanted to teach me something, but I suddenly didn’t think they had anything to teach me. I fitted the lid on my typewriter and clicked it shut. Doreen grinned. “Smart girl.”"
Later, Esther takes a bath to wash off Doreen's influence. Shortly after she becomes confused about what she really wants in life.
My research on Plath herself has remained shallow, but what I found is in line with my theory. Plath herself carefully curated her experiences to pose as such a girl in order to please her mother. (...) Evident also from early on are Plath’s all-out efforts to please mom. Describing piano practice, she writes with pride to Aurelia, “I kept saying to myself, ‘This is what mother would want me to do’“
I see her typed as 4, I couldn't have come up with that so if someone feels like explaining please do. I might feel too stuck in my 3 fix opinion to argue about it though. Edit: The only other type I could come up with was 9w8, she can be very go with the flow for example in the skiing scene, didn't seem to affect her ambition though; maybe had a 7 fix, fomo, wanting to be everything