gvim can represent multiple windows with horizontal tabs by using the -p (or --remote-tab-silent if in a single session) option:
Horizontal tabs in gvim http://void.nu/~gammy/img/tabs_hori.png
Is it possible to show the tabs in a vertical column in a GTK-fashion (ie, not using an additional vim buffer, but actually using the GUI)?
You mean – have the tab line, where the names main.pde, LevelFont.h and so on … are displayed – show in vertical fashion?
No, that is not possible.
You can use :ls to list the buffers, you can use some plugin maybe to show the list of tabs in a separate window, but you cannot change Vim’s native tabline to show differently.