Why would someone choose to work with anniversary dates instead calendar dates in the first place?

The downside on the accounting side looks real, as it’s easier to invoice all your customers at the same date (beginning of the month), instead of invoicing them granularly during a month. Imagine you want to refund a customer. If invoicing period is from 15 Sept. to 14

However, if your invoicing period is from 1st Sept. to 31st Sept., if a refund occured during this period, you make sure to calculate it during the same month. Easier for your finance guys.