While watching game of thrones (I know, not a serious source) I noticed that people always mention if they are the first of his or her name. I guess the first King Henry of England (1100) would be an equivalent. But why is it so important to address someone with that specific title?
And was that also what Henry was called from his contemporary’s?
Whilst waiting for a more direct response, there is a previous thread that may answer your question or part of it. In this thread u/Valkine explains how regnal numbers began to be used for English kings in the Late Middle Ages. According to his comments Henry I would probably just be referred to as "King Henry" during his lifetime.