What is the definition of this gaussian kernel? And why is it just integers? Note that matlab's imaging toolbox already have this function fspecial('gaussian', 3, 1.2) which returns a vastly different kernel

Only sigma, which I assume is the standard deviation is not enough to define a 2D kernel. Considering that the means for both dimensions are zero, and both standard deviations are 1.2 does not produce the exhibited kernel. A 1D gaussian curve is defined by its mean and standard deviation, and a well defined 2D gaussian kernel requires 4 parameters at least. Moreover it appears that you are cutting the gaussian kernel at its peak, the shift or height parameter should be provided as well. A 2D gaussian kernel has its peak at the center of the matrix and it does not repeat across the matrix, for instance:
0.735758882342885 1.213061319425267 0.735758882342885
1.213061319425267 2.000000000000000 1.213061319425267
0.735758882342885 1.213061319425267 0.735758882342885