When I’m saving a link using Apple Share, I would like to store it in the appropriate group. Why can’t I search the groups in my open databases? I get a bunch of recents, and then a list of others. So it ends up in my inbox where I don’t want it since I would like to get on with the next task. Frustrating.
You can’t because it’s not supported at this time.
That being said, if you’re just grabbing bookmarks there are other options available to you, like…
- Drag and drop to DEVONthink’s dock icon and use the group selector (if Preferences > Files > Import > Destination is set to Select group) There is a search function there and it remembers the last used group.
- Open the Sorter’s Navigation view and search for a group. Drag and drop the URL onto the Sorter to deposit it where you want.
- Choose Go > To Group to open the popover, then drag the popover to tear it free as a standalone panel. You can search for groups and drag and drop to it, as needed.
PS: I’m curious… why are you “saving a link using Apple Share” instead of using the browser extension?
Apple News
Well, that’s a helpful detail you didn’t mention 
I don’t use Apple News, but you can drag the article cards from the front page…
This works with the Sorter, the Go > To Group popover, and dropping directly into DEVONthink.
It does not work with the dock icon, but the Dock is also a peculiar application, so I’m less surprised. 