I don't think this is grammatically correct. I don't think there is a direct English translation that is grammatically correct. I would either translate it as "whose purple backpack is this?" or "to whom does the purple backpack belong?"
That's a shame because "Whose is the purple backpack?" is a direct and grammatical translation.
It really is a hard sentence. In this case, I'm going as parallel as possible in order to understand:
Of who (is) the purple backpack?
"Is" being invisible, but implied.
"Whose is " is no good English.
I think it is good English. The Cambridge dictionary mentions both "Whose bag is this" and "Whose is this bag" as examples.
who's purple back pack is this? this should be correct.....and yes, I know by english grammar specifications it may not be, but who speaks like that?
No, firstly who's is short for who is, which doesn't make any sense here. Secondly, Whose purple backpack is this would be של מי תיק הגב הסגול הזה?