Comment by pandaman

Visa and status are different things and exist simultaneously. A visa is used to cross the border, a status is what enables one to be in the country. Many people on H1B don't have a physical H1B visa either as they have "adjusted status" from some other status to H1B without leaving the country and hence cannot possibly have a visa, which can only be issued in a US consulate, which don't exist in the US per se.

Colloquially people refer to a status granted through a particular visa category as "visa" e.g. "I am studying on F-1 visa", "H-1B visa employees" etc.