You can suspend vim. If you try to run it in the background (e.g. with the bg command) then it just suspends itself again. To actually run it in the background on a shared terminal you need something like screen or tmux… which assigns the process its own pty.

Perhaps for most use cases involving vim it doesn't make that much difference—what would it be doing in the background anyway?—but there are other TUI programs that could be doing useful processing in the background if they didn't need to share the pty with other processes.