The second monitor is not being shared
Displays are shared one at a time. The picker shows every connected monitor as its own entry, and the one that gets shared is the one clicked — the other stays invisible even if a window is dragged onto it. For remote control that means the pointer can only be moved on the shared display.
Applications
Apple menu, top left → About This Mac — It shows the name there, for example "Sequoia 15.6". From macOS 13 onwards this window looks different than before.