I know that there is such thing as upper case letters and I know that there is such thing as lower case letters, but is there such thing as middle case letters?

Upper and lower case refer to the drawers in which the different letters were kept in the early days of printing, so I can't imagine where middle case would come into it.

https://en.wikipedia.org/wiki/Letter_case

The basic typesetter's difference between the cases:

Grand capitol -from capitol to descender line /w flourishes at other lines
Capitol -From base to capitol lines, unsloped, less ornate
Middle case -From base to ascender line, unsloped, primary body between mean and demi-lines
Lower case -From base to mean line /w ascenders & descenders; connection ready at base line, 20 degree left slope

There is also an intermediate form known as 'small caps', where the letters are in upper case shape, but sized like lower case without the tails.

Apparently, the word "minuscule" has been misspelled so often that "miniscule" is now listed in many dictionaries as an accepted variation.
Still, the corresponding word for upper case, "majuscule" is rarely seen at all.
I suppose an intermediate case could be called "midiscule" but I'm not sure what use it would have.