Probably over 99% of all 10 hole diatonic harmonicas are tuned to a standard tuning, but within the remaining 1% there's a world of variation, and it doesn't even stop at minor and minor!
If it was only natural minor, surely a standard diatonic harp would suffice?
My short answer to this question would be NO, but let me elaborate. There are several reasons.
One is the very limited set of chords on most diatonics - a standard tuned C harp has a large C major cord on all blow notes, a large G7 chord with a D note base in the bottom-half draw, and two three note Dm chords over holes 4-6 and 8-10. For playing something majory in C or G, or blues in G this is fine, but if you want chords for any kind of minor music this is rather boring.
Another is the limited set of notes on the standard tuned harp. Even if you learn to bend notes, there are two Eb, two F#, one Bb, one C# and one G# missing in the 3 octave span of the instrument. Now if you want to play in Dm, the only minor key where you have a tonic chord, it doesn't matter if it's natural minor or harmonic minor, as either the missing Bb or the missing C# will cause you trouble. The missing notes can all be produced by advanced playing techniques (called overblows and overdraws) but except for the years of practice they take to master the instrument itself must be finely calibrated way past what you find in a music store.
A third reason to use an alternate tuning is that blows, draws and bends each have a different character; in general draws and bends are considered more expressive. This is another reason D minor is a popular minor key on a C harp, because the notes D, F and A are all bendable draw notes in the middle octave. Even if we use our C harp to play blues in the key of G, the most mainstream thing imaginable, the G in the middle octave is an inexpressive blow note.
Playing the harmonica you are left with two choices: Either you embrace these difficulties and see how you can work around the limitations to shape the music after the instrument, or you shape the instrument after the music and use another tuning that is better suited for what you want to do.
Most harmonica brands offer minor key harps, usually both harmonic and natural minor. You get large minor chords, all the notes you need, and at least with the natural minor harps you get the notes in the tonic chord as expressive draw notes.
Many many pages could be written on various harmonica tunings, but I'll stop here as I think what I've written already answers your question.
Oh, and on your final question, take a look at https://www.patmissin.com/ffaq/q3.html for the history of the word 'harp' :)