Strange indeed! However,
m = {{1, 2}, {3, 4}, {5, 6}};
m
m[[2]] = Sequence[];
m = m
m[[All, 2]] = Sequence[];
m
does work! It think it is due to that m is stored like:
{{1,2},Sequence[]}
Such that there is no second element in the second element. by doing m = m you 'evaluate' the Sequence. Might be the intended behavior or not, I almost never use Sequence in this way...
I would suggest using Drop, which is intended to remove elements, columns, rows, et cetera...