The direct answer for a budgie
It is not safe for a budgie to eat food made for other kinds of pets as any regular part of its diet. A budgie is a small seed-eating bird with a fast metabolism and specific needs for the balance of energy, protein, calcium and vitamins in its food. Dog and cat foods are formulated for carnivores, rabbit and guinea pig foods for grazing mammals, and fish foods for an entirely different body again. Each of those recipes solves a different animal's problem, and none of them matches what keeps a budgie healthy when eaten beyond an accidental peck.
What actually goes wrong
The mismatch for a budgie is both nutritional and physical. Kibble for dogs and cats is hard and sized for jaws, not a budgie's beak, and its richness in fat and animal protein, plus salt levels birds tolerate poorly, burdens a small bird's liver and kidneys when raids become routine. Vitamin and mineral premixes in other pets' foods are dosed for animals many times a budgie's weight, so repeated eating skews nutrients the bird also gets from its own diet. Wet and raw pet foods add a second hazard: bacteria that a dog's gut shrugs off can sicken a budgie, and food left in another pet's bowl has been standing, licked and contaminated. Even other birds' foods are not automatically suitable; a large parrot mix is mostly seeds too big and too oily for a budgie.
If your budgie has already eaten some
A budgie that has pecked a little dry kibble or stolen from the rabbit's bowl once will usually come to no harm, so do not panic, but do move the other pet's food out of reach and offer the budgie its normal food and fresh water. Watch over the next day for the signs that matter in a bird this small: droppings turning watery or changing colour, food coming back up, unusual sleepiness, or sitting fluffed with eyes half closed. Any of those after eating another pet's food, or any raid on wet, raw or spoiled food, is a reason to contact an avian veterinarian promptly rather than wait, because budgies deteriorate quickly once they stop eating normally.
Setting up a mixed-pet home safely
Feed the budgie in its cage or on its own stand, and feed dogs and cats behind a closed door or at set mealtimes with bowls lifted afterwards, so there is nothing to raid between meals. Budgies on free-flight time explore by beak and will investigate a cat's bowl left on the floor, so the bowl, not the bird's curiosity, is what you manage. Store every animal's food in separate sealed containers with separate scoops, which stops cross-contamination and accidental mixing at serving time too. Keep the budgie's own offerings interesting, with varied vegetables and foraging toys, and a well-fed, entertained bird bothers other bowls far less.
What belongs in the budgie's bowl instead
A budgie does best on its own measured seed or pellet base with daily vegetables such as leafy greens, carrot and broccoli, plus clean water changed at least daily. That combination covers the nutrients the other-pet foods get wrong for this species, in portions a budgie's body is built to handle. Treats marketed for budgies and small birds are portioned for them in a way a dog biscuit never is. When you are standing in the kitchen wondering whether a particular shared snack or stolen mouthful matters, KinsLe's Kinny can be asked for guidance in the moment, and what you learn can stay alongside your budgie's profile for the next time it happens.