Cardano adrestia links to several different options for interacting with Cardano from 3rd party applications & languages. cardano-wallet provides a REST API to manage a wallet, and can send/manage tokens, but does not appear to have minting functionality just yet.
AFAIK, the only way to mint tokens at the moment (if you don't want to use Haskell SDK) is by using cardano-cli as documented here. Creating tokens dynamically would then require writing a layer on top of the CLI, since several intermediate steps are involved in transaction building.
For the client-side of your game, you may want to consider 3rd party APIs like blockfrost.io for fetching user's NFT data, or provide it from your own server along with regular game data.