Translation:I need a pillow and a blanket.
In Brazil, pillow is travesseiro.
Almofada is pad. (Found this translation via google, and I'm not confortable with it) So, there's some pictures of "almofadas" to leave no doubt at the end of this comment. They are used on sofas and chairs. Can be part of the sofa, or additional ones put onto it.
So, one cannot say a "travesseiro" is not some kind of "almofada", but we never call it that way.
To me, a non-native speaker, the list constitutes what I called a "noun phrase" (though now I'm not really sure that term applies to a list) and one "de" should be sufficient.
It is not always appropriate to take the rules from one language and apply them to another, but I guess it is similar to the difference between saying "I am fond of reading and of writing" and "I am fond of reading and writing".
I think your comparison to the English sentences works. "Eu preciso de um travesseiro e de um cobertor" sort of separates the two items, almost as if you needed them for two different things or the blanket had come in as an afterthought. "Eu preciso de um travesseiro e um cobertor", on the other hand, makes a block, a unit of "blanket + pillow". Both are correct and it is impossible chose one best translation from the English sentence they propose.