This is an English translation from "Garante" which is actually a stronger word - more like Guarantor. It is an official authority with teeth.

Certainly in UK English we use watchdog to mean any organisation that has an oversight role, frequently government ones. For example the Financial Services Authority might be described as “the banking watchdog”, it is very much a government agency.

Why do you think watchdogs have to be non-governmental?

For example:

https://www.theguardian.com/technology/2022/may/05/uk-watchd...

It's likely a bad translation.

The Italian SA is the Italian Data Protection Agency (DPA), one of the per-country European regulators https://ec.europa.eu/justice/article-29/structure/data-prote... . Which acts under the GDPR and predecessor data protection laws, and is very explicitly a governmental regulator.