Greek definitely has upper and lowercase. Arabic and Persian do as well, but also rules on glyphs (different letter-shapes based on proximate letter combinations), as well as small accents that change the pronunciations.
All have hand-written (cursive or calligraphy) vs typewritten variations, including fixed-width vs variable fonts.
Serifs are display attributes, mostly for latin alphabets. But greek lettering have had them too, albeit in subtle, light versions. The modern didone (thick) or slab serifs didn't show up until 18th Century.
You're right. For Arabic and Persian, was trying to simplify the idea of an intitial, medial, final, and isolated form and map them to latin upper and lowercase.
For those interested, the same letter can take different shapes depending on where it appears in a word. For example, the same letter 'H' can be: