A pair of scales.
Most of the riddle is just describing them, and the last lines allude to their purpose.
Two twins we are, and, let it not surprise,
Alike in every feature, shape, and size:
We’re square, or round, of brass or iron made,
Sometimes of wood, yet useful found in trade;
Scales may be made of various different materials, they consist of two identical parts, and they might be often used by old-time traders.
But, to conclude, for all our daily pains,
We by the neck are often hung in chains.
Either we're equal or one is greater.
They dangle side by side from chains, and whether they're equal or one is greater is the important thing in using them.