They are lying. Anything you do digitally is logged. The data to identify you is there. Therefore, anyone wishing to exert power over you will access that data if they have the means (which they must do, as someone at the organisation has a way of retrieving it), irrespective of whether that is intended or permitted by the organisation or by law.

There is no such thing as anonymity.

I once translated evidence in an unfair dismissal case - at a mining company in Chile of all places - and the boss had found a simple way to identify who had given what feedback simply by cross-referencing the data as to age band, department, gender, etc. Even without getting anyone from IT in to trace it back to each individual.