You need to click and hold the mouse, and drag down to the Abandon or Quit, then release the mouse button.

Oh wow, okay. Thanks. I guess that makes sense for touchscreen, but maybe add a toggle to the options which is off by default on platforms which are not android/ios? Or have it popup on hover under those OS?

edit: To clarify, i didn't even see the popup menu which is shown on hold, since it is visible for such a short amount of time on a normal click.

The interface is actually designed for mouse use, but it isn't very modern, true. But just changing it to require two clicks to select any items from a drop-down menu will also make the interface worse, since several of the main toolbar buttons depend on having a single-click "smart" choice. Especially the Build Rail and Build Road buttons let you choose between different rail and road types (when they are available), but a click on just the button will select the last used type. Similarly, you typically save your game between 3 and 10 times more often than you leave it, so it makes sense to have it on a single click action. But there might be other ways to make it more discoverable for new players without forcing old players to break their habits.

Makes sense. Maybe something like an info popup on the very first launch of the game?

I've been playing a bit online lately, and indeed the problem recurred quite a bit.
Currently it is not a problem, but I didn't like this "press, hold and select" solution for quite a long time. It is not popular solution, and therefore it is also not intuitive. OTTD is probably the only game/program I know of with such a solution. In fact, wherever there are additional options hidden under an icon or button, pressing it opens a new window or list. So for some new players it can be troublesome.

Perhaps the solution could be to add an option to speed dial a feature. When turned on, the game would behave as it does today. When disabled, pressing the icon would open the list and require an additional click - it would work just like selecting the order. Except, pressing anywhere else would close the list.

Another option would be to change the concept where pressing an icon would always open a list or an additional icon bar.

I think that something like this would be even more convenient than the current solution.

Working as intended