What is the name of the feature that allows copying on one Apple device and pasting on another?

The feature that lets you copy on one Apple device and paste on another is Universal Clipboard. It transfers copied text, images, photos, videos, and some other content between nearby Apple devices signed in to the same Apple Account.

Universal Clipboard is part of Apple’s Continuity framework. It uses a combination of Bluetooth for discovering nearby devices and Wi-Fi for transferring the clipboard data. Both devices generally need Wi-Fi and Bluetooth enabled, and Handoff must be turned on. The copied content is temporary: it remains available for a short time or until a newer copy operation replaces it.

Universal Clipboard is different from Handoff, although the two features are related. Handoff continues an activity inside a compatible app, such as a Safari page or an unfinished email. Universal Clipboard transfers the selected clipboard contents themselves. It is also different from AirDrop, which is intended for deliberately sending saved files or other items. Apple introduced Universal Clipboard with its modern Continuity features in macOS Sierra and iOS 10.