The broken "More" button for a game in the games library seems to be because the other options are not included with the span element that contains the "Assign Tags..." and "Changlog" options. When I move the other options into that span element they then appear okay.
Using Safari 11.1.1 on macOS 10.13.5, I'm able to use the inspector (Control-click on the "More" text, select "Inspect Element") to reveal its element:
<span class="module-header2__dropdown game-details__dropdown--capitalize _dropdown is-contracted" gog-dropdown="">More
...
It helps to change the "is-contracted" class name to "is-expanded". Just double-click on the text "contracted" and it will be highlighted for editing. Press return when done changing the name. The styles will take effect and show the menu without needing to click on "More", and without having to keep pointer hovered over the menu.
Inside that element is the element containing the drop down arrow, and after that is the element containing the "Assign Tags..." and "Changelog" options:
<span class="_dropdown__items game-details__dropdown-items js-init-scroll">...
To see the elements inside, open that element by clicking on the triangle next to it. The two options are inside, along with the menu for the "Assign Tags...", but ignore those because they appear okay in the page.
Anyway, having it opened also allows for easily dragging and dropping other elements into it. The menu options happen to be anchor "a" elements that are listed after the menu, immediately after the ending of the span element: </span>. For example, the first is the link to the "Store page & reviews":
<a class="_dropdown__item _dropdown__item--contrast" ng-show="details.url.storePage" ng-href="/game/myst_masterpiece_edition" href="/game/myst_masterpiece_edition">Store page & reviews</a>
Click somewhere on that "a" element and drag it upwards towards the </span>. A thin horizontal line will appear indicating where the dragged element will be inserted when the letting go. So move just before the </span> and let go. It'll immediately appear after the "Changelog" option in the menu on the webpage itself, and will look normal like the other two options. It also turns out that once it is done for one game, it stays correct for all games. Well, until the webpage is reloaded.
Since the options look normal when put into that span element, it seems to me they are being accidentally left out of that span element. That's just a big superficial guess because there is way too much markup and CSS and Javascript for me to ever try to know anything for sure.
A client-side fix would be to push those other options inside, probably with a little bit of Javascript, but I figure the server-side will be fixed sooner than it's worth to do that (other than as an exercise, or racing the GOG webmasters).