Kindof both questions of why - what's the technical implementation of the special death animation and why go to the effort of making a unique animation for robot enemies and not for other enemy types like say humans or birds.
Also depending on what is found in the code, maybe that'll shed light on the more philosophical 'why' question. It could be that there are stubs or half-written flags to represent death animations for other classes of enemies but it was never finished.
As to the Red XIII being called "Red" https://youtu.be/XokJ1e_rRSE?list=PLBO7HwXkx1urqa2nqM38Jcbzbaj0tPR7M&t=298 It's the last dialog box before you get to input a name for him, he says "Hojo named me Red XIII but you can call me whatever you like" the box is captioned as being said by "Red", not "Red XIII" or just a nameless text box but "Red". My theory on this is that they kept changing the characters used for his name and forgot to correct it in this one instance. In a pre-release screenshot I found on FFWikia it has the name spelled out in arabic numerals as "Red 13" https://vignette.wikia.nocookie.net/finalfantasy/images/b/b0/Red13-prerelease-ffvii.jpg/revision/latest?cb=20120824222843 but in the released Japanese version there's a single character for the number 'XIII' just to use for naming Red XIII https://youtu.be/jlVFrktbcG0?t=321 Then for the English release they spelled it out in full as "Red XIII". The majority of the time anyone uses his name it'll be referencing the variable that you can change when you rename him, but not this one box before you name him and evidently they got it mixed up and ended up not giving any numbers at all.