Exactly, this is the main confusing point for those that use a 24 hour clock. A digital clock using that format will count the hours of the day and at 23:59 will got to 00:00 of the next day.
Most Europeans perfectly understand that AM is night/morning and PM is noon/evening. But the tricky part is how the times are displayed at noon and midnight. For Example midnight:
24hr clock: 23:59:59 -> 00:00:00 -> 00:00:01 etc.
12hr clock: 11:59:59 PM -> 12:00:00 AM -> 12:00:01 AM ->> 12:59:59 AM -> 01:00:00 AM
This is why many Europeans are confused whether its released midday or midnight. Personally I this the second example looks weird too, but if you are used to it its totally natural. Btw im not even 100% sure on that one so correct me in neccesary.