I haven't used neovim, but it certainly is possible to use vim with its terminal feature as a terminal multiplexer. The way I do it is to use a single instance of vim to handle the multiplexing and just run whatever I need in different split windows or tabs.

If I need to copy, I'll use the terminal normal mode (ctrl-w N) and visually highlight text and copy it to the + register. Then I can paste the contents in another application. The only issue I've encountered is that some text that's not supposed to be wrapped has line breaks added at the screen width. In that case, I'll paste the text into another vim buffer first and remove the line breaks using gJ. then copy and paste it into another application.

For copying text from another application back into vim, I'll just highlight it with the mouse and paste it into vim using ctrl-w "*

I started using it before that was introduced. I haven't tried going back.