As Tim said, 6/8 is complicated, and there are more questions on it here. I would note that in general a tempo indication should indicate what durational value in the notation that it is referencing. In 6/8, that's usually a dotted quarter (or dotted crotchet, for those outside the U.S.). However, I've definitely seen tempo indications for 6/8 given for the eighth note or even (very rarely) for the dotted half.
That is, at the top of your score, there should be some indication saying, "Dotted quarter = 128" (usually a little version of a dotted quarter note, rather than writing the name out). Or "eighth note = 128" or whatever.
As for the reason that the dotted quarter is almost always seen as the primary "beat" in 6/8, it's because of a whole history of music notation that we don't need to get into right now. Suffice it to say that in music, from the earliest days of rhythmic notation, it seems groupings of beats into 2s and 3s were most popular. And subdivisions of beats into 2s and 3s were also the most popular. (4s could also be seen as twice 2, but 5s and higher numbers weren't part of the groupings.)
Hence, we have a system of time signatures where the beats are divided into two parts (so-called "simple time"), like 2/4, 3/4, and 4/4, which have 2, 3, and 4 beats per measure, and where each quarter note beat can be broken into two eighth notes.
And we have a system where the beats are divided into three parts (so-called "compound time"), like 6/8, 9/8, and 12/8, which have 2, 3, and 4 beats per measure, and each dotted quarter note beat can be broken into THREE eighth notes.
All other standard time signatures generally contain only these six numerators (2, 3, 4 or 6, 9, 12), with a different denominator to indicate the relative size of the pulses.
In sum, please do NOT think of a time signature as "the top number is the number of beats and the bottom number is the note that gets the beat." That's wrong, despite the fact that many music beginners are taught that, due to teachers attempting to oversimplify things. For time signatures with a top number of 5 or bigger (particularly the standard numbers of 6, 9, and 12), there are usually some other groupings of smaller note values that are felt as the "beat."
So why would you even bother to change the denominator if the timing
stay exactly the same, but just the notation of the notes change?
That's a separate question: why would you write 2/4 with quarter note tempo at 128, but you could also write 2/2 with half note at 128 or 2/8 with eighth note at 128? The brief answer is that yes, they are all basically equivalent notations, so there's some redundancy. (The longer answer has to do with historical practice: certain styles of music tended to be notated in 2/2, while others were in 2/4 or 4/4, etc. But most of those distinctions are less important in modern music.)
As for the difference between 3/4 and 6/8, the basic difference is explained above, but there are a bunch of other questions here on that exact topic. It has to do with whether you group the eighth notes into 3 beats of 2 eighths, or group them into 2 beats of 3 eighth notes.