Every charlatan has an angle he or she uses in preference to other schemes
I shave coins or forge documents.
Is there a formula or mechanic for shaving coins, or is it more of a flavor thing?
There are no built-in mechanics specific to the schemes you can choose when you have a charlatan background. But if a player likes the idea of one of them and wants it to come up in play rather than just being a part of their backstory, then a GM can probably figure out some mechanics to make it work.
One idea with some official rules support is to make it a downtime activity. Chapter 2 of Xanathar's Guide to Everything has some ideas that are very close to what you'd want. If a character wanted spend their downtime making money by shaving lots of coins or forging paper money, they could probably use a variation on the Crime activity, which can earn between 50 and 1000 gp in a week depending on how much challenge and risk the player is willing to take on. The amount of payout determines the DC for a number of ability checks using different skill or tool proficiencies. You could pretty easily modify the thefts described in the book to be a forgery or coin clipping operation with the same potential payouts and many of the same complications, but using a few different skills than usual (e.g. forgery tools, rather than thieves' tools, and maybe deception would be mandatory and stealth optional).
A DM could also just make up their own home-brewed mechanics for what the player wants to do. A simple approach might be to just give the character a discount of a few percent on anything they buy with shaved coins (since they get to keep a small part of each coin's value in the shavings). But any time they do this, the person they're paying gets a Perception check to notice that something is fishy with the money (the DC might be something like 20 minus the percentage discount). If the shopkeeper rolls well, they might demand extra payment (perhaps even beyond the discount the player received), or call for the city guard!