A breather does just that, breathes. Attached to the axle is good. Other end is just open so it can breath. Keeps the axle from having positive or negative pressure inside. Makes up for thermal expansion.
From the factory no filter is used. And really not needed either. There is extrememly little air flow through the breather. The hose is long enough that dust just sticks to the inside wall and stays there forever. The open end is simply pointed down so stuff doesn't fall into the hose.
In the realm of making things better, lots of people add hose length to raise the breather. Higher location generally makes it harder to dunk the breather underwater. Added length also adds more filtering as there is more hose to collect dust. And if you really want to spend as much as possible you can add little K&N filters, or the cheap way out of just putting a fuel filter on.
There are millions of cars and trucks running around that have no breather hose at all. Just a vent on the axle itself. Either by design, or the hose rotted off. This works just fine by all accounts. Probably half the old junk you find in a junkyard will be this way, and usually with a perfectly good axle in it as well.