YouTuber rhotoscopic stabilized United 93 footage from the 2006 film, illustrating why shaky-cam is an often-used creative choice.
Director Paul Greengrass faced several challenges with the film. Everyone knew how it ended, and there are two very different locations where most of the action takes place. In order to visually tie together the chaotic and claustrophobic footage on the plane with the footage of people calmly talking on phones and looking at monitors, shaky cam helps make the on-the-ground footage seem less ordinary. Without the shaky cam, the actors often seem as if they are just having a busy day at an office.
It also shows some of the "rules" around shaky cam use. The simpler the composition, the more freedom to shake the shot. Several shots mimic human eye movement, following hands to inanimate objects. Most shaky cam footage also has an upper threshold of dutching, or angle for turning the camera. Greengrass also doesn't use many snap zooms in this sequence, which typically make viewers more aware of the camera and feel more like a documentary.
Update: here's the original (in wrong aspect ratio) for comparison:
