Does interest and tax earned get added to the bank balance/ banker's account balance or does it go somewhere else ?

If you are referring to the taxes applied for payments sent, the amount a Banker sets their fees at will be added to the banks funds. There are also national fees and If I am not mistaken, they are removed from the game as a gold sink.

Interest I would naturally assume you mean the interest you would receive from a loan contract, which is added to the bankers balance.

By banks funds do you mean the banker's account in his own bank ?

Yes

Bank fees go to the Bank Balance.
Nation fees (tax for loans) go to Nation treasury (a way to fund your nation treasury directly).
There are also corruption fees, which are removed from the game.