It looks like the normal dolphin UI, but I cannot figure out how to enable a search bar for this specific pick file window. Is it even possible? I’ll specify I am using Wayland here, since this is a third party Wayland client.
So this is desktop-portal. Specifically xdg-desktop-portal-kde.
Depending on what you mean there is no search bar.
You can type into the selection area and the selection will jump to the typed sequence.
You can type into the path above that area as well.
If you want to actually search, such as using the index or similar to query a matching file in the whole system - then you can search separately using dolphin or kfind etc.
It is a common and unfortunately stubborn misconception among many newbies that the desktop portal — i.e. the file chooser — of KDE Plasma would somehow be dolphin, just as the same amount of newbies keep on mistaking kscreenlocker for sddm with equal persistence.