FAQ

General answer - Well, people do still use them, and it wasn't very prominent in the film, so it may not have any deeper meaning. But, Peter isn't shown to be a very trendy kid ever, and most of his parents' resources seem to go toward managing Charlie and other family difficulties. It's not hard to imagine them forgetting about his concerns even if he did ask for a new monitor- and his mother's focus on Charlie is key to allowing events to unfold as they do.

Tech answer - 4:3/ 5:4 monitors offer more effective screen per inch than 16:9/ 16:10 monitors do in many cases. Much information is presented vertically, rather than horizontal. Try reading a text-book with pages vertically wider than broad. Few widescreen monitors offer a proper Pivot mode, useful for spreadsheets and stock-trading. For graphics-editing classic-ratio is also often preferred. As 4:3/ 5:4 monitors are relatively cheap on the secondary market (even those with good color reproduction for professional graphics - not affordable for most otherwise), they are used beyond the above mentioned reasons by people with lower budgets (possibly like a teenager in a movie; you may as well ask why he's not using a phone or tablet - or a laptop). 16:9 or 16:10 is only relevant for visual active/ interactive content like movies or gaming. In fact I still have my old high-end CRT (tube!) perfect color reproduction monitor shipped out to me once in a while from storage when working on a high-end project (cheaper than buying a new $ 24,000.- near-perfect color-reproduction flat-screen).

Cinematographic answer - Having (closeups) of 4:3/ 5:4 monitors on a 16:9/ 21:9 (or otherwise widescreen) ratio film provides directors and cinema-photographers with "scope". For closeups one can either alternate between background and foreground (in focus) in the same frame, or use atmospheric background (lighting/ color/ movement while remaining focused on the screen. This setup also helps to draw attention to the center of the screen, making displayed content easier to see for the audience - photos/ text (our physical field of vision is quite narrow, note that when you read this, everything is slightly out of focus, except for the few letters that you are reading).

There's no reason to assume Peter was always disconnected, so it's likely his book is in a similar state to his mental health because of how hard his life has become.

He is a therapist and in a deleted scene, offered to help Peter talk through the loss of his sister.