Specifically, this is father's younger brother.
Technically speaking, yes. But it's also a generic form of address for an older male relative.
"Today is the birthday of my uncle" should also be accepted.
I think Duo, in any course, prefers standard translations. Though "today is the birthday of my uncle" is a literal translation of the sentence, it is awkward and unnatural.
my uncle's birthday is today should be accepted
"Today's my uncle's birthday" should be accepted too
I wrote this too