You're missing a fundamental idea, and as soon as you understand it, everything will come clear. The idea you're missing is the idea of the
smart contract. The
wallet you mention is a
external account and can hold ether. It has a public/private key pair. You personally hold the private key, and only you can sign messages (i.e. make transactions) on that private key that allow the ether to be moved around.
Another kind of account, and the kind that you need to grok, is called a smart contract. A smart contract is an account, in the sense that it can also hold and spend ether, but it's different than an 'external' account. No-one owns the private key of a smart contract. No-one can control it directly other than by sending it transactions (function calls) from an 'external' account.
Smart contracts, in addition to holding ether, and different from external accounts, also hold
state. The idea of
state is what you're looking for in answer to your question.
In your case, you can 'store' the points in a smart contract's state. You program smart contracts with a programming language called Solidity. One of the constructs in Solidity is a map. You can store the points in a map indexed by your user's accounts. One deploys a smart contract by sending a special transaction carrying the compiled Solidity code from an external account.
Not to get too complicated, but you'll probably come to realize, after time, that storing a lot of state on the Ethereum network can be astronomically expensive, and you'll seek out other solutions such as storing the actual state data on IPFS and storing the only the IPFS hash on the smart contract, but that's an issue for a later date.