When you cut a bar magnet in half carefully, it will create two smaller magnets, each with their own north and south poles. The dual polarity of magnets is part of their fundamental nature. It's analogous to breaking apart two Legos; each will still have a top and a bottom.
Magnetized objects contain grains of magnetized material throughout their structure. The way these grains are aligned creates a magnetic field with two poles (north and south), regardless of their size or orientation. If you were to connect two bar magnets together end to end by their attracting sides, you would essentially have one magnet with one magnetic field.