When I have this problem this indicates to me that someone needs to assign descriptions to the items in question (and I've done that on several instances).

I think this problem would be better resolved by hooking into the statements on an item first (P31/P279) as with the auto-description gadget, which already fills in on the dropdowns (so far as I recall it does so).

In my interface, which prefers Czech, descriptions usually fallback to English (or Slovak)...

In T173035#3517136, @matej_suchanek wrote:

In my interface, which prefers Czech, descriptions usually fallback to English (or Slovak)...

MediaWiki-defined fallbacks work, but user-specific fallbacks (from Babel boxes) don't. That seems to be how all Wikibase-specific parts of the interface work.

re-tested; some behavior has changed for the better, but some issues remain (see updated task description)

It would seem that the suggester now searches labels in all languages, but when searching by item ID (Q…), if there is no label for the UI language, there is still no fallback to any of the user’s other languages. While this might be difficult to implement (requiring analysis of Babel boxes or the like), an easier intermediate solution might be to fall back to official name (P1448) of the suggested item.