Do yacht charters start on the same day every week

No, not usually. Usually, it’s just a matter of where the gaps are in the yacht’s booking calendar. The only exceptions are:
1) Christmas and New Year’s weeks. Boats always book both these weeks, so they have a specific change over date from the Christmas charter to the New Year’s charter. Yachts may also want ten nights minimum over this period. They don’t do under a 7-night trip.