As Tim already mentioned in the comments, it could generally be called accompaniment.
You can categorize these in different types like:
- mixtures of multiples
Each of these can be categorized further like for example a baseline could be:
- walking baseline
- alberti bass
I think how these are typically build, depends a lot on the genre of the music. But they will all have in common, that they add some harmonic and rhythmic context to the melody.
Generally speaking, you could try to analyze the underlying harmony of your song/melody and use these harmonic information to build one of the types of accompaniment I have listed above.
Of course some theory knowledge will help you to analyze the harmonic information and choose how to make use of it.
Your question seems specifically pointed towards music in the style of Bach. I'm not very experienced in this type of music, so I will let this open to other users which might have a better grasp of explaining how an accompaniment could be build in this style of music.
As for the question if the left hand is counterpoint, Wikipedia says
In music, counterpoint is the relationship between two or more musical lines (or voices) which are harmonically interdependent yet independent in rhythm and melodic contour
So one could say, that you could play musical lines with your left hand which will create a counterpoint with your melody in the right hand, but your left hand playing will not necessarily be counterpoint.