There's a thing called "constructive receipt" that comes into play.
You have paid for a service in 2022 that will not be performed/provided until 2023. Effectively, you have permanently lost control of that money in 2022. (since you no longer have it.) Therefore, the medical expense paid is claimed on your 2022 tax return. This would still hold true even if you paid with a credit card for which you will not receive the bill for, until 2023.
Likewise, the recipient of that money is the one who has "constructive receipt", meaning that have complete control of that money. Therefore, it is reportable by the recipient on their own 2022 tax return, since they received the money in 2022 and can use that money any way they please.
