$$yi+xj$$ Curl of above vector field is $1-1=0$. I thought this means if we put an object in the vector field, it wouldn't rotate. However if I put an object on $y$ axis at say $(0,5)$, it will rotate clockwise because the velocity is increasing with $y$. What am I missing here?
Curl (like grad and div) are local properties which means that if you put a small cross in a fluid without curl then it will not rotate about its centre.
On the other hand the global motion of the cross (which moves with fixed orientation) in the fluid is pretty much unconstrained by the vanishing curl so the centre of the cross can definitely move around in lines that encircle the origin.
See https://www.falstad.com/vector/ and select the 2d visualisation with curl crosses to see the point.