handheld game console; portable game console
Refers to the hardware device itself, such as a Nintendo Switch or PlayStation Vita. Often used in contrast to home consoles.
See also: 携帯型ゲーム機
Handheld game consoles are convenient for playing during commutes.
handheld game software; game for a portable console
Refers to the game software or title designed for a handheld device. The distinction from the hardware sense is usually clear from context.
I downloaded a new handheld game.
Compound of 携帯型 (portable type) + ゲーム (game). A straightforward modern term.