The usage is not just limited to east but also in northern india and I know it's also used in telugu. In my understanding of the usage (north india), it can refer to father or as a sign of respect. It was also used to refer to the clerical staff under British Raj and even now you can see that usage.
To add to Khaanabadosh, name+babu (yes, following our usual HU transliteration, it would be "baabuu") is the common formula of respect in Eastern India. However, it is exclusively a male designation.
In Bengali and Bihari, it is also used stand-alone (i.e. without any name attached) as an affectionate term of address to younger people, e.g. parents to children (at least in Bengali, it is acceptable to use this also towards girls, though probably less common than towards boys). In this, it is similar to Hindi beTaa. I don't know if the equivalent use exists in Oriya, Assamese, etc.
In Bengali, I have never heard "babu" being used for father. However, I could find this meaning listed in a Bengali dictionary. So, presumably it existed at some point. In any case, baba > babu transformation would be parallel to what has happened to many other Bengali terms of family, kaka > kaku, chacha > chachu, mama > mamu, dada > dadu, etc. So, historically speaking, it makes perfect sense that babu started off as "father". However, in my contemporary usage, it does not exist.