What is this oversized playing card deck for, with numbers 11-15 and Barons and Knights?
The deck wasn't made with 20 cards in each suit to match a particular kind of game — it was made with 20 cards because that's the number of amino acids.
Google translate gives us some indicators of that from the product description:

4 to one of the base 4 in one of the suits, 20 of amino acid 1 from 20 (...)
Normal playing cards 13 compared to the sheets, because the number [ed: of amino acids] is large, J (Jack, 18 th), Q ( Queen, 19 number ) , K (King, 20 th) in front of the 16 as the number K N (Knight, Knight) , 17 as No. B are (Bishop, priest) the picture card of the addition.

Though very rough (very rough), they're acknowledging playing cards are usually 13 to a suit, but they voluntarily expanded the number and added a couple of picture cards because there's a larger number of amino acids than just 13.
The page consistently says it's for Trump (the Japanese name for our standard 52-card playing cards) and never identifies any other game for which you'd use these.
Instead, it's something brand new, as they close off saying, and it's for you to figure out what you might like to do with these cards:

Because it is a new playing cards, to devise how to play, while having fun, the root of the living world is a principle DNA should understand the relationship between the amino acid.