Switzerland sits outside the EU but inside the Schengen area, and it has a long-standing veterinary agreement with the EU that covers pet travel. In practice, this means an EU pet passport (or a Northern Ireland one) is accepted for entering Switzerland exactly as it would be for entering an EU country, and Switzerland issues its own Swiss pet passport, which the EU treats as equivalent for travel between Switzerland and the EU.
Like the EU version, the Swiss pet passport is only issued to pets whose owner is a Swiss resident, by a Swiss vet. If you live outside both the EU and Switzerland — the US, UK, Canada, Australia and most other countries — neither passport applies to you. You'll need an Animal Health Certificate instead, arranged through an accredited vet before you travel.
One detail worth planning around: if your trip includes both an EU country and Switzerland, check whether one certificate covers both legs or whether Switzerland requires its own separately endorsed paperwork, since the US-to-Switzerland export process is handled as its own listing by USDA APHIS, distinct from the general EU one.