I am not sure what you mean by

There is now multiple ways to represent this same exact game state in my network because each individual card has it's own state. To me this does not seem good.

We can think of game state as a column vector of the form 1xn. In this case, you can formalize cards held according to some encoding. This can be anything from a Gödel Numbering to simple integer assignment. Here is an example for poker in python: https://pypi.org/project/treys/

In the example of duplicates you would have a column vector that simply has duplicate indices. I.e: {241, 424, 112, 112, 455}

I am not sure what you mean by

There is now multiple ways to represent this same exact game state in my network because each individual card has it's own state. To me this does not seem good.

We can think of game state as a column vector of the form $1Xn$. In this case, you can formalize cards held according to some encoding. This can be anything from a Gödel Numbering to simple integer assignment. Here is an example for poker in python: https://pypi.org/project/treys/

In the example of duplicates you would have a column vector that simply has duplicate indices. I.e: {241, 424, 112, 112, 455}$