Usage of 'customs' in lieu of 'immigration'

When entering the United States, first you go through immigration, where they check your passport (and visa if you are not a citizen or permanent resident). Then you go through customs where they check your bags and other items. In many airports these two processes are handled in the same room, so referring to one by the name of the other should not be surprising. Also, since customs comes last, it is the indicator that the whole process of going through regulatory control before being allowed into the country is complete, so saying "clearance through customs" implies clearance through immigration too.

NOAD includes people as one of the things checked through customs (travelers emphasis mine):

customs |ˈkəstəmz| |ˈkəstəmz|
plural noun
the official department that administers and collects the duties levied by a government on imported goods : cocaine seizures by customs have risen this year
| [as adj. ] a customs officer.
• the place at a port, airport, or frontier where officials check incoming goods, travelers, or luggage : arriving refugees were whisked through customs.
• (usu. customs duties) the duties levied by a government on imported goods.

So I would say that this is not a mistake, but common usage—at least in the U.S.