How do we know for sure that vector addition works while working in 2D?Is there a mathematical proof for this?
The parallogram rule for the addition of forces was described by Isaac Newton. You can test it with wires and calibrated spring scales in any undergraduate physics lab. Or use pucks on an air table.
For mathematical proof, see the first chapter of any linear algebra text for a full treatment.
The key is to know when a physical quantity is a vector: forces, velocities, accelerations, etc. Then you can rely on the math.