In England, at least, those were the names they were born with. Just like now, sons were named after fathers or grandfathers (or other more distant relatives) and they kept those names when they came to the throne. William the Conqueror had two sons, William Rufus and Henry. William Rufus became William II when he took the throne, and after his death his brother became Henry I. Henry had one legitimate son, named William, who died before he could come to the throne; his grandson was named after him and became Henry II. The name Edward was given to Henry III's son because Henry was particularly devoted to Edward the Confessor, an earlier English monarch and saint, and then that name got passed down. Richard II may have been named after Richard I (the Lionhearted); his father Edward (known as the Black Prince) was a highly respected soldier who had already named his eldest son Edward, though both father and son died and Richard wound up taking the throne after the death of his grandfather. It's possibly worth noting that the name Richard first made its way into the list of monarchs out of chance - he had an older brother named Henry who died before he could take the throne. Same with John, another younger son who has a terribly poor reputation (heavily debated whether that's deserved or not), and thus hasn't had his name passed down to any other potential monarch.

I don't know when choosing a regnal name that was different from the birth name became a thing in England, that's beyond the period I'm most familiar with, but from the Anglo-Saxon period up through at least the beginning of the 1600s (when Elizabeth died) it was just what they were named when they were born. Doing so had much the same goal it has now: honoring an immediate relative, calling to mind a prestigious ancestor, or drawing connections between the individual and a prominent religious figure.