It's not that weird a term: I was taught this as the Roman name for the Mediterranean in middle school history class in Spain, without having to take Latin. There's a boardgame and a video game with Mare Nostrum as the title. I's expect relatively well educated people in countries bordering the Mediterranean to understand what he meant with little trouble, especially if they are also speak a latin-derived language. For instance in Spanish, mare is just mar, nostrum would be nuestro, and mortuum is muerto. It'd all be trivial first guesses.