How come a Slavic country such as Montenegro got a name that sounds so Italian?

It's because the name that has caught on is Italian! The name of the country in its native language is Crna Gora.

"According to the Constitution (ustav) of the Republic of Montenegro, in Montenegro the Serbian language of the jekavian dialect is the official language. Unlike in Serbia, Cyrillic and Latin alphabets are deemed to be equal. In the municipalities in which the majority or a substantial number of the population consists of the national minorities and ethnic groups, their respective languages and alphabet are in official use."

http://www.montenet.org/

Because it used to be part of Venice.

Montenegro is actually named "Crna Gora" in Serbian, which means "black mountain." But the Republic of Venice controlled the area in the Middle Ages (including the Dalmatian Coast in neighboring Croatia), which is where the Italian name comes from.

https://en.wikipedia.org/wiki/Republic_of_Venice
https://en.wikipedia.org/wiki/Montenegro

It comes from the the Italian "monte negro" meaning black mountain. This is also the meaning of the country's native name, which is Crna Gora.
The name Montenegro itself reflects the time when Venice ruled over that region. The reference to black mountains in turn comes from the excessively mountainous geography of the country.

"Montenegro's native name, Crna Gora ("Black Mountain"), is mentioned for the first time in 1276 by Serbian King Stefan Uroš I (1243-1276) in his edicts to the Serbian Orthodox Zeta Episcopate seat at the Vranjina island in Lake Skadar. The origin of the term lies in the Slavic reference to excessively mountainous regions, often emerging in the medieval Serbian realm."

"The country's name in most Western European languages, including English, reflects an adoption of the Venetian term monte negro, meaning "black mountain", which probably dates back to the era of Venetian hegemony over the area in the Middle Ages."
https://en.wikipedia.org/wiki/Montenegro

The name "Montenegro" is not Italian but rather Venetian. "Black" means "nero" in Italian, not "negro" as in Spanish or Venetian. As a matter of fact, the country used to be called "Montenero" in Italian.

https://en.wikipedia.org/wiki/Montenegro#Etymology