Is it permissible to claim private car expenses (equivalent to the taxi fare) when it is used for maintaining the Islamic bank's ATMs, knowing that this work is not part of the basic duties and is not compensated?

The reference for travel compensation is the agreement with the bank. If the amount for travel is absolute, then it is yours. However, if the agreement is to compensate only what was spent on taxis, then it is not permissible to take more than the amount spent. You can explain your use of your private car to the bank; if they agree to give you the equivalent of the taxi fare, then there is no harm.