Thanks, that looks nice. However this:

set pumborder=rounded
or
set pumborder=+,-,+,|,+,-,+,|

That looks inconsistent (using a single word vs. several single characters). Also I think would make sense to model it after the popup borderchars option (for popups). We could then later re-use this option for the popup borders as well, so perhaps call it just borderchars option then.

The right border wall should be shifted one cell to the right, placing the scrollbar inside the border—consistent with the behavior of popup_create(). In fact, all other aspects should align with popup_create(), including support for border, padding, borderchars, borderhighlight, and so on. There’s no justification for introducing inconsistent/confusing special case UI like "rounded". Instead use borderchars as intended.

That looks inconsistent (using a single word vs. several single characters)

single word is some built-in border types. Neovim has some built-in border types. This provides some convenience, borderchar is not easy to port to neovim. I implemented winborder option in neovim to set float window (vim is popup window) border, but it is only used for windows, not pum. Pum and popup are different things. Is it appropriate to use popup fields?

I think this feature is not worth adding. The code for computing popupmenu position and size is already quite complicated, and adding a border to the popupmenu will even further complicate the code and introduce more edge cases.

Not entirely correct. When the height is sufficient, we add an extra border. When the screen space is insufficient, the upper edge and width of the current pum are additionally occupied. code is not complicated

Also, mouse select position is currently off-by-one in right-click menu with 'pumborder' enabled:

source $VIMRUNTIME/menu.vim
set mousemodel=popup
set pumborder=rounded

fixed .