Whenever you drag an anchor of a paragraph-character-anchored object, related object moves together with the anchor. This makes it impossible to fine-tune anchoring of objects on screen, when their positioning is set.
1. In a Writer document, add several paragraphs.
2. Insert an image, and anchor it to the first paragraph.
3. Now try to drag the *anchor* of the image, with the goal to re-anchor it to paragraph 2, but not to change its position (assuming we want the image to be linked to the second paragraph's text, but be positioned at the same place where it was before).
=> The image moves to the following paragraphs when you drag the anchor, so the only way to make the desired change is to enter numbers into the Properties dialog.
Of course, the proposed method (bindable to some keyboard modifier like Ctrl+Alt+Shift) should limit the movement of the object to disallow making the object and the anchor on different pages.
This is the reverse to bug 141161.
I confirm the described situation with
Version: 184.108.40.206.alpha0+ (x64) / LibreOffice Community
Build ID: f96004096268f5e71120678e32fc8c74055819aa
CPU threads: 4; OS: Windows 10.0 Build 19042; UI render: Skia/Raster; VCL: win
Locale: de-DE (de_DE); UI: en-GB
but I think this is an enhancement request rather than a bug.
For me one general problem is, that LO has only very short informations about image anchoring  and we have a lot of report about it.
Wouldn't implement this as another anchoring method. Moving the anchor with alt being pressed, for example, should do the trick.
New enhancement since the use case is clear and reasonable.