Base it on Tournaments
You have some institution where wealthy people gather—perhaps a casino, but maybe a church, university or social club—develop and sponsor the game, including commissioning the official cards and rulebooks. The objects might not be cards as we understand them; etched wood blocks are a good possibility.
If you want to enter the tournament, you need to buy (or rent) the official game pieces made by the tournament, and since all of them were made there by hand, the officials can spot fakes. If you want a copy of the rules ahead of time, so you can make a strategy or practice with dummy cards? You have to pay for it. There might or might not be rules about what you can have in your deck other than how much money you are willing to pay the house to win the tournament.
When the printing press arrives (Cai Lun’s printing press without movable type was well over a thousand years old at the equivalent point in OTL.) it will be possible to make the backs of the cards exactly the same, so they are not all marked. If those are not in use, the rules probably have to make the game an open game, where all cards are known to all players at all times. An alternative is to keep hidden cards behind screens and have servants manage the decks.
Sell it to Gamblers
Some cards are deliberately rare and powerful, and for the right price, might be sold or even given out to the right customer so he can be sure he will always have it to play, not his opponent.
At that point, the house might start keeping the game fresh by limiting what cards are available and issuing new themes. (If you paid them for the right to bring a card to any game, you probably keep it, but maybe there are exceptions.) Other competitors might copy the game (with their own made-up cards and rules).
As with the other tournaments, the house knows what artist made each of its cards and whom it sold them to, so it can easily spot forgeries. Or it might simply keep the cards in its vault and give them to you whenever you walk in.
You don’t want to let the same people buy too much of an advantage, or else others will stop coming to play with them, but social stratification means that aristocrats will want to play with each other, not the riff-raff, and will feel the need to own a respectable deck. Especially dedicated players who invest a large amount of money in their decks get their own table to play each other, but sometimes others accept a game with them—at least they’ll get to see the unique and expensive handmade cards. (The house surely has a policy that new players get to start with good cards, so their first experience with the game isn’t miserable.) If someone powerful gets some great cards and wins over and over, that at least keeps them happy with your establishment. Besides, as long as they come in the door and pay to try out your latest cards for a week, or buy their lucky card, the house wins regardless of who takes home the pot.
An example of a rule meant to ensure the house always takes its cut: if you win the pot with a card the house lent you, you either pay the house rent on it, or you may buy the card on the spot.
Play it at Home
People might start commissioning their own cards and playing each other at home, If social roles are anything like the real world, this is especially how women would play, so perhaps the game played by noblewomen entertaining guests evolves into something different from the game played by gamblers in casinos or men in tournaments. (In your world, social roles on how noblewomen are allowed to socially mix might of course be looser.) An aristocrat would not want a reputation of being a poor sport with her guests! And often, you want the person you’re entertaining to win. So, the home version has an incentive to keep a level playing field. This game probably lays out for display all the beautiful, expensive cards the hosts had made for them, has all players select an equal number of cards from those, and gives the guest player the first pick of them. In the common case that two married couples play each other, lady guest first, then hostess, than gentleman guest, then host, round-robin. The servants then arrange the decks and shuffle them.
Maybe it Catches on
If there’s a reasonably-standard set of rules, it might be possible for people to make and bring their own cards to play each other; you can’t just draw your own Black Lotus because everyone knows what the real cards are. It might also be possible to show your decks first and have the players or a judge decide whether they are reasonably fair. Otherwise, the model the game probably follows is the guest-host convention: the “host” brings enough cards for all players to build a deck, and the guest gets first pick. Perhaps a competitive match involves each player taking turns as the host.
An important part of the strategy, as the guest, is quickly analyzing the cards the host presents and deciding which to choose and which are traps. The tradition of being a good host and a good sport remains important—as a way to trick the marks. Sharks love tricks like copying a well-known card with small changes to the wording that the guest won’t notice until you point them out in play. And it’s a mistake to agree to play anyone who’ll lawyer any card in the game unless you have some way to force him to accept a fair ruling.