There's an inherent difficulty in getting a screenshot to both file and clipboard. The screenshot app does one or the other, not both, so to have both you must actually allow it to finish saving the file to Desktop then open it in something you can copy to clipboard from, such as Preview.
A slightly quicker method, which may at first seem counter-intuitive, is that if you select the file on the Desktop & copy it, the Finder quite cleverly copies all aspects of it & then can paste different aspects of the file, depending on where you paste to. Paste to a folder it will paste the file itself; to a web page it will paste the link to the file; to a text app it will paste the file's title; and to an image editor it will paste the image itself.
Clicking the screenshot floating thumbnail doesn't actually open it in the Preview app. It opens in a version of the QuickLook app [with extra annotation, similar but not identical to Preview]
QuickLook
You can from here click the Share icon, top right & send it to Preview.
Preview
If you want the initial floating thumbnail to go to the copy buffer, right click it before it vanishes & select Save to Clipboard.
This, however, will redirect it entirely & it will no longer save to desktop as well.
You could at this point send it straight to Preview app [which would lose it from the clipboard - there's no win/win here ;)
All of these choices are the same whether you use Cmd ⌘ Shift ⇧ 3 or Cmd ⌘ Shift ⇧ 4.
From either of these, adding Ctrl ⌃ to the keyboard shortcut will send the image directly to Clipboard rather than file [yet again an either/or decision]
Cmd ⌘ Shift ⇧ 5 has its own prefs which you can change once it opens, rather than by adding Ctrl ⌃
You could actually eliminate the floating thumbnail entirely if you wished - this also makes the save to Desktop a bit faster. You do this by using the Options menu in Cmd ⌘ Shift ⇧ 5
If this all sounds a bit confusing, it's because these commands have kind of grown organically over many years, as new features were added. Once you understand them you can see the relationship they all have to each other, especially if you think of them as being oldest to newest, 3, 4, 5.
The newest Cmd ⌘ Shift ⇧ 5 implementation has the most control over what you get from it - including fullscreen, window, selection snapshots as either stills or movies - with power comes complexity… but still has no way to send to both Clipboard & file in one shot.