the singer is Johnny Preston ...
I like the term "bathroom baritone" but usually in the US that kind of singer is called a "crooner."
I've heard the song lots of times. I don't find it offensive, or especially memorable. That style of sappy, sentimental, storytelling ballad is very dated. I would almost call it a novelty song, except novelty songs are more often than not humorous. (Think "The Streak" or "A Boy Named Suee" or, God help us all, "The Purple People-Eater.")
Many Indians did indeed have names like Running Bear or Little White Dove, or at least their Indian-language names translated into those. So, yeah, the names are cliched, but I wouldn't call them or the song offensive; traditionally, names were chosen to reflect traits of the person, or traits it was desired that the person would possess, as well as circumstances of that person's life or birth. And often people acquired new names at different stages of life. And, face it, it was desirable for males to have certain characteristics and for females to have certain others. That's just how it was.
Sappy, dated, and obviously yet another reworking of the whole Romeo-and-Juliet story of star-crossed lovers.
It probably wouldn't get produced today, not so much because it is offensive as just because it is so far out of step with current musical styles.
(Johnny himself probably didn't write it. I should look it up, but my bet is it was churned out by career songwriters and he happened to be the one who got to sing it or else his version was way more popular than anyone else's.)
Before you criticize a man, walk a mile in his shoes. That way, when you criticize him, you're a mile away and you have his shoes!
Steve Martin