Short answer
Yes, mussels can give some cats diarrhea, especially if they eat too much or have a sensitive stomach. While cooked mussels (shells removed) are generally safe for cats, any new food can upset their digestion. Offer just a tiny piece first, and if your cat has loose stools or seems uncomfortable afterward, skip the mussels and call your own vet for advice.
Mussels — Food safety facts
Safe to share · Cats · full breakdown on the linked food page
Cooked and removed from the shell, in small amounts. Raw shellfish carries bacteria.
The full verdict, portion guidance and serving notes are collected on the Mussels food facts page.