We are new to Shopify and had to do our 1st refund this week. Customer was initially charged €44.35. Refund was due for €3.45 which we did through the system. However customer has come back to say his 1st charge of €44.35 is still charged and he was charged €40.90 (difference 3.45). He has got no refund but was charged again.
Definitely a bit of a strange situation, but I think I know the answer to why they are seeing two charges. Did the charge and the refund take place on the same day? If yes, then this would be a case of a "reversal of funds" which means the refund won't appear on your customers' banking app/statement. If you refunded €3.45, then the correct charge would be €40.90; however, the second "charge" of €44.35 comes from the initial authorized amount. Authorized charges are not full charges, they are only the amount the system authorized to be charged. As you refunded the order and changed the total charge within the same day, your customer is seeing both the authorized amount and the actual charge as banking apps don't differentiate between the two. Authorized charges typically fall off within 2-5 business days as per the processing time of your customer's bank. Your customer can also call their bank directly who should be able to provide the same information.
If the order wasn't refunded on the same day, then let me know as I can verify the store via email to take a deeper look.