The first brother sings and sings, but never runs out of breath,
The second brother runs and runs, but never tires,
The third brother eats and eats, but is never full,
The fourth brother drinks and drinks but is never sated,
Who are they?
Could the four brothers be
The Empedoclesian elements? (Earth, Air, Fire, Water)
The first brother sings and sings, but never runs out of breath,
This one is Air in the form of wind. The singing refers to the wind whistling.
The second brother runs and runs, but never tires,
This one is Water in the form of a stream or river. Rivers run on and on.
The third brother eats and eats, but is never full,
This one is Fire, as fire engulfs anything in its way.
The fourth brother drinks and drinks but is never sated,
This is Earth, as dirt and soil soak up water (From OP's comment).
Old Answer
If the other answers are right, that would make this one Earth. I'm not really sure as to how Earth "drinks". Perhaps it's talking about plants needing water to survive?
My guess is
The parts of steam locomotive
Reason The first brother sings and sings, but never runs out of breath,
You are referring to the chimney which blows the whistle if you can create music by blowing it
The second brother runs and runs, but never tires,
This is the heart i.e the Engine that always runs but is never tired
The third brother eats and eats, but is never full,
From wikipedia https://en.wikipedia.org/wiki/Steam_locomotive These locomotives are fueled by burning combustible material—usually coal, wood, or oil—to produce steam in a boiler.
The fourth brother drinks and drinks but is never sated,
This is a guess but by this you are referring to the boiler that is used inside the steam engine. From wikipedia https://en.wikipedia.org/wiki/Boiler
To sum it up
Together they are the steam engine
I think these are
Products by the company Brother (in the company logo the b is not capitalised).
The brother who sings and sings is
A sewing machine (who does what Singers do)
The brother who runs and runs is
Perhaps a gear motor
The brother who eats and eats is
A scanner (eats paper)
And the one who drinks and drinks:
An inkjet printer (always runs out of ink)
The first brother sings and sings but never runs out of breath would be Conquest or Glory. The second brother runs and runs but never tires would be War. The third brother eats and eats but is never full would be Famine. The fourth brother drinks and drinks but is never sated would be Death. (Vampiric)