As in most of Europe, Swiss taxi drivers aren't under a legal obligation to accept pets, and it's left to individual discretion. In practice, most are relaxed about a well-behaved, leashed dog or a cat in a carrier, especially if you mention it when booking rather than surprising the driver at pickup.
Uber and similar rideshare apps operate in Switzerland's larger cities — Zurich, Geneva, Basel, Bern, Lausanne — and work the same way as elsewhere: there's no guaranteed pet-friendly option, so a polite note in the app or a quick message to the driver ahead of the ride goes a long way.
A travel mat or blanket for the seat is a small courtesy that Swiss drivers, like most drivers anywhere, tend to appreciate.