Very basic question about filters in signal reconstruction
Neither a filter with a rectangular impulse response, nor one with a triangular impulse response is a very good interpolation / reconstruction filter. If you use a filter with a rectangular impulse response, the resulting output is just a piecewise constant signal. Since you want the constant pieces to have exactly the duration between two samples, the width of the rectangle must equal $T_s$. You could call that a zeroth order interpolation.

A first-order or linear interpolation between the sample values is achieved with a triangular filter kernel. If you look at it in the time domain you can see that the triangle with its apex aligned with a sampling instant $k$ must extend from the previous sample $k-1$ to the next sample $k+1$ in order to provide linear interpolation when all those triangles are summed up. So for linear interpolation, the base of the triangle must have a width of $2T_s$.

Of course, that triangular function is the convolution of two rectangles of width $T_s$, so its Fourier transform is the square of the Fourier transform of the rectangular pulse.