1. Was the katana really that old?
Yes, it was - we have both Ramirez's words and Dr. Wyatt's test results. She says,
Brenda: I dated these pieces of the blade at 600 B.C. The metal's been folded over 200 times. The Japanese didn't start making swords that way until the Middle Ages. So where the hell did it come from? If I could verify its existence, it would be like discovering a 747, made a thousand years before the Wright Brothers ever flew.
(from the script - the dialogue spoken in the film is slightly shortened, and transcripts available online have some mistakes)
So the blacksmith had to be a real genius, that made a one-of-a-kind sword.
The Highlander wiki says:
The preferred weapon of Immortal Connor MacLeod, the ivory-handled Masamune katana is a weapon like no other. It was forged in 593 B.C. by the legendary master swordsmith for Immortal Juan Sánchez Villa-Lobos Ramírez, as a dowry for his marriage to Princess Shakiko. Using revolutionary techniques which would not become common until the 14th century A.D., Masamune folded the metal in the blade over two hundred times, creating the katana's infamous razor-sharp edge.
Ramirez himself mentions it to McLeod:
Ramirez: MacLeod, I was born 2,437 years ago. In that time, I've had three wives. The last was Shikiko, a Japanese princess. Her father, Masamune, a genius, made this for me
in 593 B.C. It is the only one of its kind... like his daughter.
So it is not a screenwriter's mistake, but a purposeful plot device.
2. Is the Masamune who made the sword the same famous blacksmith from the 14th century?
That's possible, but very doubtful: Masamune from Ramirez's story had a daughter, and immortals cannot have children. Also, why would he hide for over two thousand years, not making a single sword, just to resurface again in the 14th century?
The out-of-universe answer would suggest that the screenwriter simply took a famous name to look more believable, but from the in-story point of view it's possible that the 14th century Masamune simply found the original Masamune's notes, and was able to use them to create new swords. Possibly, to honor the original blacksmith, he took his name.