Grammatically fine, "on my period" has the form of a prepositional phrase "preposition + (determiner+noun)".
Idiomatically correct too. Idiom is determined by use and this is the phrase that is used. Whereas "on my flow of blood" is not idiomatic. As Lambie says "They are idiomatic expressions. You can't just delete and replace these terms with "flow of blood". That is an explanation; not a synonym."
You could investigate how this idiom developed over time. It would have to be a fairly recent idiom since "period" in the sense of menstruation dates back to 1830. The word "period" refers to a length of time, which points to a likely development of sense:
We do tend to say "On Tuesday" or "On the weekend" with a period of time lasting a day or a few days. So "On my period of menstruation" would seem to be a natural generalisation, which would be shortened to "On my period".