"This book is given to Andi."
Translation:Buku ini diberikan kepada Andi.
You're right, these words have (subtle and not so subtle) differences in meaning.
'dikasihkan' has the base word 'kasih'.
'kasih' can be a verb or a noun, and in this case the verb (not the noun) is the base word.
This is the KBBI definition:
kasih2/ka·sih/ v cak
siapa yang -- kue ini?;
As you can see, it's a synonym of "beri", "memberi".
However, as you can also see there is this term 'cak' in the dictionary.
'cak' (ragam cakapan, tidak baku) indicates that it's a word used in the colloquial/everyday speech (ragam cakapan), and that it's not the standard (tidak baku).
Although it's a synonym and it's perfectly fine to use it like that in everyday speech, it's not used in the 'Duolingo' translation.
So the main difference is the formality.
'dibagikan' has the base word 'bagi'.
'bagi' can be a particle or a noun, and in this case the noun (not the particle) is the base word.
'bagi' = part of a whole.
KBBI never lists the 'di-verbs', so we have to look at the active verb:
memberikan (kepada banyak orang):
Ibu - kue kepada anak-anak;
As you can see, the synonym is 'memberikan' (the active form of "diberikan").
However, this verb has a different meaning.
It has the meaning that you are giving 'parts (of a whole)' to several persons.
'memberikan kepada banyak orang = to give to many persons.
(you are sharing the object)
The KBBI example sentence :
"Ibu membagikan/membagi-bagikan/memberikan kue kepada anak-anak."
"Mother is giving the cake to the children."
"Mother is giving the cake (in parts/slices/pieces) to the children."
So, if you use this verb with a book, it could mean that you have torn the book apart in pieces and you give Andi one piece of the book and another person another part of the book.
Not really how you would like to receive a book.
I think it's better to use "diberikan".
Yeah, it makes sense. Rick articulated the differences quite well. I guess it makes sense with "bagian" being a part of something, but I'm just connecting all the pieces in my head. In my mind, I associated it with sharing something, "bagi bagi" as you said. But then my English brain gets confused, because I think "well, I enjoy sharing my books with people or sharing my thoughts" - though I suppose in that sense you're not really physically sharing anything between multiple people at a time.