What is the ruling on zakat for funds deposited in Islamic banks as investments, given that the state annually collects zakat from expatriates?

Zakat (definition) is obligatory on deposited funds if they reach the nisab (definition) and a full year (hawl) passes over them, whether by themselves or by adding them to other funds. If the state takes an amount equal to the obligatory Zakat and you intend it as Zakat, then you have fulfilled the obligation. If the amount taken by the state is less than what is due, you must complete the remainder. If it is more, and you can reclaim the excess, then do so. Otherwise, seek your reward from Allah.