To answer your question directly, no. A PdS is solely for residence in Italy. It is completely moot (except for short term travel purposes) in other EU/EEA countries. Each has its own residence permit,(*) and none of them are called permesso di soggiorno.
(*) The U.K. is one exception. A residence permit is not strictly required (since the U.K. doesn't have a residence registration system for its own citizens), though it's recommended.
Correction: The U.K. EEA family permit (a visa, effectively) is neither required nor recommended if the foreign spouse already enjoys visa free access to the U.K. and if the EU spouse is not a U.K. citizen.