I used the answer of NM Pennypacker and installed Vim via Homebrew for an early 2011 MacBook Pro:
brew install vim
Now I can also use the "* register to copy and paste text within Vim. I even didn't have to change something within my ~/.vimrc file or the $PATH. Homebrew added symlinks in /usr/local/bin, e.g., vim -> ../Cellar/vim/8.1.2350/bin/vim.
The alternative, which worked before, is to copy some lines of text within Vim by marking it with the mouse and using copy and paste (Cmd+C, Cmd+V) on a Mac. This option only works if the text you want to copy and paste is less in size than the window size of Vim. If you want to copy all text within Vim by marking the whole window or using Cmd+A, this will copy other parts of the console, written before starting Vim, which is very annoying.
So, I am happy having found the new method using the clipboard register.