Familiar cards are *not* respected when you play a rated game. So if you go to the "Matching" tab and under the "Automatch" section either click on "Two player" or "Three player", you try to enter the queue for rated games and when you find opponent(s), you enter a rated game and nobody's Familiar Cards are respected.
In order to have your Familiar Cards respected you need to click on "Practice game" (for a game against another human player) or choose 1-3 AI bots as opponents under the "Bot game" section in the "Matching" tab.
There are various advanced options you could change but the basic rules is the same: If you play a rated game, Familiar Cards are not respected. Of course, if the opponent you find via the automatch for rated games has (like I assume is true for you) no subscription to an expansion, then the game will select a kingdom on the combined set of cards that both of you are subscribed to, which would only be the base set because that is free for everybody.