Is expiation (kaffarah) obligatory for one who broke his oath, while not remembering the type or wording of the oath, and is it permissible to give the monetary value of the expiation to charity funds?

The basic principle is that whoever doubts whether they uttered an oath or not, or whether they swore one oath or multiple oaths, then they are considered not to have uttered it, or that it was only one oath. This is because the original state is the acquittal of responsibility, and doubt does not remove certainty. An oath is not binding merely by intention.

If you are certain that you uttered the oath, then if it was a single oath concerning multiple things, its expiation is one. However, if you swore independent oaths for each thing, then an expiation is due for each oath upon breaking it, contrary to the Hanafis and Hanbalis in one opinion that allows for multiple expiations.

As for placing the expiation in a donation box like that, it is not sufficient because there is no certainty that it will be spent in the legally prescribed channels for expiation.