If I've eaten more than one bread/piece of bread, I say that I ate "mine brød", whereas if I only ate one or an amount that can't be neatly quantified (like part of a loaf), I say that I ate "mit brød". Same holds true for everyone I know. The word "bread" itself may not have a plural, but sin/sine is still based on whether there is one or more of the item.
Same goes for sheep. Et får (one sheep), to får (two sheep). Manden bar sit får ("the man carried his (one) sheep"). Manden bar sine får ("the man carried his (multiple) sheep").