You can do this. Turn on pane status lines (set `pane-border-status` option to `top` or `bottom`). You can then change `pane-border-format` to contain the text you want. It can show the application-set pane title (`#{pane_title}`, also settable with `selectp -T`) or anything else really.

There are no builtin pane names but if you are running tmux 3.0a or later you can just set a pane user option like `set -p @myname name1` then use it in `pane-border-format` with `#{@myname}`.