How is soil measured? By ton, pounds or etc.

Really depends where you are.
In the UK it tends to be measured by volume, square yards.

USA-(New Jersey)

In American roadbuilding it is measured in cubic yards. In some cases, it is delivered with a certified weight ticket and then has to be converted to cubic yards which requires that you know pounds per cubic foot. Cubic yards is the most prevalent.
But some states may require by the pound or ton (with a weight ticket) to protect the consumer in the case of topsoil type materials.

Usually by volume, but at times by the pound. Depends on the amount being purchased sometimes as well. It could be by the truck load, which could be volume, or tons. Then it could be in a bag, such as potting soil, that could be sold by the pound.

Soil can be measured in pounds, tons, kilogram

When you get, or order a dump truck load, at least in the US, it is in Cubic Yards, about 2,000+/- # per cubic yd.