These changes make everything much more intuitive and streamlined. I will explain the system as a whole in more detail below.

Let's start with Silver medals. By default, you have a Silver medal on each card you own. When you play a match vs AI, this means you bring all your cards with a 1x multiplier. Each card then has an equal chance to get selected for your Shop, and the same for the AI's Shop.
In PvP it works similarly, except now there are 2 sets of medals that get combined, your set and the enemy's set. An example:
Let's say you have only the Base game and your enemy has DLC. Then a Base set card, like Tanya, will have 2 medals and thus a 2x multiplier. A DLC card, like Scientist, will have 1 medal and thus a 1x multiplier. This means for both of your Shops, Tanya is 2x more likely to be selected than Scientist.
OK, those are the basics. If you understand how that works, the only extra thing is that there are different types of medals that have different multipliers. Gold medals are the next type of medal you will get.

Where Silver medals have a 1x multiplier, Gold Medals have a 5x multiplier, which is a big step up. Let's start with how you get them. Every time you win a match, you will get a Gold medal for a Base set card, so eventually you will have all 40 of them. For the DLC, there is a mission for each card that you can play to collect the Gold medal for that card. You can find these missions in the Collect menu. If you only play PvP, you will get them after you collected all the Base set ones.
Now that you have both Silver medals and Gold medals, you can choose for each card which one you want to use.
I would say there are 3 main ways to use your medals:
To round up, there are two more types of medals: Super medals and Ban medals.
Super medals give you even more control over what shows up in your matches, with a multiplier of 15x.

Ban medals can be used on cards you really dislike, with a multiplier of -14x you will basically never see them in your matches. You will get these medals as rewards for leveling up.

I hope that helps! If you have any questions, feel free to ask in the Steam community, or on Discord.