Animals want to stay with their owners. This is why there are different types of accommodation for refugees with pets where they can take their pets with them.
Here, you can find accommodation offers from private individuals where you can take your pet with you: Private accommodation at TASSO
The regulations of the federal asylum centres are cantonal and must be clarified in each individual case: Cantonal federal asylum centres
If you cannot be accommodated together with your pet, in many areas the animals are first placed in an animal shelter. Swiss animal shelters are of a very high standard and in most cases, you can visit your pet during its stay. Video of a visiting day at the animal shelter Pfötli VIDEO