The problem is in the way you wrote your last equation as a matrix multiplication.
You have $$ [ \textbf u ( \nabla \textbf u) ]_i = u_j (\partial_i u_j) = (\partial_i u_j) u_j, $$$$ [ \textbf u \cdot ( \nabla \textbf u) ]_i = u_j (\partial_i u_j) = (\partial_i u_j) u_j, $$ so if you want to write this in matrix form you have to multiply the vector $\textbf u$ at the right, as a column vector, i.e.
$$[\mathbf{u}\cdot(\nabla\mathbf{u})]=\left(\begin{array}{cc} \partial_{x}u_{x} & \partial_{x}u_{y}\\ \partial_{y}u_{x} & \partial_{y}u_{y} \end{array}\right) \left(\begin{array}{c} u_{x}\\ u_{y} \end{array}\right) $$$$[\mathbf{u}\cdot(\nabla\mathbf{u})]=\left(\begin{array}{cc} \partial_{x}u_{x} & \partial_{x}u_{y}\\ \partial_{y}u_{x} & \partial_{y}u_{y} \end{array}\right) \left(\begin{array}{c} u_{x}\\ u_{y} \end{array}\right), $$ where the derivatives are intended to act only on the adjacent $u_i$.
Anyway, I don't see where the covariant derivatives comes in here. The $\nabla$ you are using is simply a gradient, not a covariant derivative.