"Her cat eats everything."
Translation:Её кошка ест всё.
Essentially, "У неё кошка ест всё (or всё ест)" is a sentence about HER, not about her cat. Sentences with disconnected "у + someone" dangling around the start, express some situation relevant for that person (or that object):
- У меня заболел брат. = My brother fell ill (so that's why I skipped a class)
- У двери отвалилась ручка. = The door's handle has broken off. (now it's hard to use that door)
- У неё пропал голос. = She lost her voice. (so she cannot sing anymore)
In all of these examples, the situation ("brother fell ill", "handle broke off", "voice got lost") is assigned to a person or object for which it MATTERS.
English just uses possessive, i.e., it is "My brother fell ill" regardless of how you feel about it (whether you feel involved or it is more of your brother's problem).