I at first read this as:
"Tiwari, Rai was the one responsible for appointing unit presidents for BJP's Delhi unit and Bihar unit".
However, you're quite correct that it could mean that
"Tiwari, Rai was appointed to be unit president of both BJP's Delhi unit and BJP's Bihar unit"
Where Tiwari, Rai is an individual; the construction would indicate that this person's first name was Rai and their last name was Tiwari.
"Tiwari and Rai were appointed to be presidents of BJPs Delhi unit and Bihar unit respectively
Where Tiwari and Rai are different people.
In this context, I would lean towards the last interpretation; it would be rather odd to break up a person's name into Last, First structure in the middle of a sentence, so it's more likely those are the names of two separate individuals.
Context would be needed to clarify what was meant here, unfortunately - the sentence itself is ambiguous.