More or less. It depends on the animal, the canton and what exactly the clause says. Because Switzerland has a fairly elaborate jurisprudence on this, and the real situation is considerably more nuanced than many owners imply when they send you the contract.
For small animals that do not produce any verifiable harm, the blanket ban is not legally upheld. A hamster, a bird in a cage, fish in a fish tank: the Swiss courts have made it quite consistently clear that prohibiting them from entering, when they do not cause real damage to the property or discomfort to the neighbors, exceeds what the lease law allows. So even if the contract says “forbidden animals,” that point doesn't apply without nuance.
Dogs and cats are another category. There the owner has a real margin. If the clause expressly prohibits pets or requires prior authorization, it must be requested in writing. Some Romanian cantons have a jurisprudence that tends towards permissiveness when the animal is quiet and the floor has exterior access, but that is not the federal norm and there are differences from canton to canton.
What is constant is what happens when you leave: if you had an animal without authorization, the owner can claim expenses for deep cleaning, carpet or rug treatment, or any damage attributable to the animal, directly from the bond. There are people who discover it at the time of return, which is the worst time to discover anything related to renting in Switzerland.