Inspired by that episode analysis video, I thought it would be fun to do some armchair psychoanalysis of the characters of the show.
Twilight Sparkle has obsessive compulsive tendencies that sometimes reach almost self-destructive proportions. She wants to be in control of every single detail in her life and have everything planned to the utmost detail in advance, and she is constantly seeking knowledge that, ostensibly, is often more focused on controlling every single aspect of her life than anything else. She hates not understanding things (such as Pinkie's inexplicable sixth sense) because that means there's something she's can't handle. She's almost morbidly afraid of disappointing her mentor and may go to extreme lengths to avoid it. She was very introverted as a filly, and still is to some degree (but has got much better thanks to her friends.)
This tells me that she has some deeply-seated insecurities. Deep inside she's scared of not being in full control of her life and the events surrounding her. She's scared of leaving things up to chance and improvisation instead of being carefully pre-planned. It's unclear whether this is because of some uncaused neurosis, or if it's perhaps caused by some traumatic childhood event (maybe parental abandonment or other type of traumatic experience.)
Pinkie Pie seems to have similar issues, but they manifest themselves differently. Pinkie is almost pathologically afraid of being alone, of being abandoned by her friends, and sometimes this reaches outright paranoid levels. She tries to be friends with everybody, even obsessively so. She has difficult time accepting the fact that perhaps someone does not necessarily want to be friends with her. She always wants to show a happy face to everybody, but sometimes this is a facade, and instead she's just hiding the fact that she may be hurting inside. She often has difficult time showing her true feelings to others, when she's not happy. She's an extreme extrovert, even to a fault.
This tells me that she suffered some very serious childhood issues, possibly related to her parents and family. She probably felt extremely lonely and unloved as a filly (her parents being very uptight and rigid, after all.) There might have been some even more traumatic events. Now she's trying to compensate these deeply-ingrained traumas by trying to make sure that everybody is her friend. The more friends, the better. It's a psychological defense mechanism against the morbid fear she has of being alone and unloved. She's trying to combat loneliness by going to the other extreme. All this tells me that she still has some unresolved psychological issues that unfortunately nobody around her has noticed nor tried to help her with. (In a more realistic setting Pinkie would be, in my opinion, a worryingly risky candidate for suicide, something that would greatly surprise everybody around her.)
Rarity, besides being somewhat obsessed with cleanliness, seems to also have a mild case of inferiority complex. She thinks of the upper society class as something to aspire towards, but something she can probably never reach herself. Her passion for fashion and cleanliness might or might not be a product of this complex. The fact that she has a fashion boutique in Ponyville, which is extremely rural in comparison to Canterlot, might or might not be an indication that she doesn't feel worthy of living in the latter, if not even a sign of she subconsciously seeing Ponyville as a place where she represents the upper class, where other ponies around her would look up to her as a dignified high-cultured important pony. This is probably mostly subconscious, though, and not something she consciously thinks. Regardless, she's very kind and generous to others, rather than feeling superior and detached.
This might be the product of her upbringing, if her parents were very strict and put heavy expectations on her, expectations that she felt never could be completely fulfilled.
For a very long time, and since early childhood, Fluttershy was painfully shy, timid and introverted. She might have been bullied as a filly. She feels much more comfortable taking care of her pets than with other ponies. At some point she's almost afraid of other ponies, especially strangers. Much like Pinkie, if she feels that even animals don't want to be her friends, she can get extremely neurotic and paranoid, even aggressive. She has gotten significantly better in this regard thanks to her friends, but obviously she still shows signs of this. It's unclear whether her timidity is innate or whether it's the result of traumatic experiences as a filly. (I'd guess it's the former, although the possible bullying certainly didn't help.)
