Giant system dialog font on Windows 10
@Yorik's comment pointed me in the right direction.
The problem is, indeed, with the fonts used for MS Shell Dlg which, by default, is Microsoft Sans Serif. The latter is described as "metrically compatible with the MS Sans bitmap font" which hints at why high-DPI scaling can get messed up.
Solution: Locate
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\FontSubstitutes
in the registry and change MS Shell Dlg to something else - Arial or Tahoma will work, but to look like the rest of Windows I chose Segoe UI.
Dialogs are now back to normal.