Hi, let's say it's 2020, mining with my current ASIC doesn't yield anything, but I supposedly can be rewarded for verifying the blockchain by fees.
How does a transaction (which is not my) get processed through my node and I get a fee?
Or is the fee system only for exchanges and the likes who keep accounts for others?
You don't have to wait until 2020. You are already getting fees when mining (or at least the pool you are using is, depending on the pool it may or may not be sharing those fees with you).
That's what mining is: getting paid for transaction processing and securing the blockchain.
Every block that is solved includes both a subsidy (currently 25 BTC, it will be 12.5 BTC in 2020) and transaction fees of every transaction that is included in the block.
The miner (or mining pool) chooses the addresses that the payment will be sent to before they start mining the block.