What exactly is the Masamune sword?

The name Masamune doesn't reference an actual sword, but a legendary sword smith called Gorō Nyūdō Masamune. (Altough, his name is also attached to his swords, like Honjo Masamune, Fudo Masamune, Hōchō Masamune and Kotegiri Masamune.) And for this to make sense, you need to understand that Masamune isn't regarded only a great swordsmith, but the great swordsmith. You can read more from the linked Wikipedia-article.

The same hold true for his pupil, Senji Muramasa, whose name you've no doubt also heard in similar contexts. (Pupil as supposed by legends, actually, as some sources state that they didn't even live at the same time periods.)

Masamune's and Muramasa's names are widely used in Japanese popular culture to refer to either legendary swordsmiths or swords. So much so, that this has became a sort of cultural meme.

Something I see often is western game developers, who most likely know the names only through games, using these to refer to only swords, as opposed to people.

For more on the cultural side of the two names, mostly their use on swords, see their TV Tropes article (beware of getting lost on that site).

It is likely because of the Japanese origin/inspiration of many games.

"Masa" means approximately elegant, splendid or genuine (or perhaps military) from here

and

"Mune" means approximately ridge or heart from here

There are other games that utilize the -mune suffix for swords as well (ex: Manamune in League of Legends)

Further, there is a Japanese swordsmith Masamune who is commonly regarded as Japan's greatest swordsmith. His name has been used in many Cultural References