For example, I docked the Export dialog on the left

However, after I close it and show it again (Shift+Ctrl+E), it reappears on the right:

How could I make it stay where it was?

Maybe this:

Thanks for the reply, but that option has always been on for me. And Inkscape still refuses to remember where the dialog was docked.

Not reproduced in 1.4.2, Win11, nor Linux Mint 22.1.

Please:

1. Describe your OS (your computer's Operating System: Mac/Windows/Linux/etc. & version).
2. Describe if Inkscape was newly installed, or upgraded from an earlier version.
3. Describe the version of Inkscape and the installer you used (the file you downloaded from the web: .msi/.exe/.zip/appimage/other, PPA/Snap/Flatpak, dmg/other ). 
4. Describe any special hardware being used, like tablet/stylus, external drives or multiple monitors.
5. Describe any helper/assistive programs or keyboard modifiers (macros, languages, etc.).

- OS: Windows 10 22H2

- Inkscape Version: Inkscape 1.3 (0e150ed6c4, 2023-07-21) (newly installed, not upgraded from an earlier version)

- Special Hardware: No

- Assistive Programs: No

Please check the attached file for a screen recording showing steps to reproduce this bug.

Also, weirdly, if I use floating dialog, the position and size are properly remembered by Inkscape. But docked ones' are not.

Ah. I see better now.

Reproduced in Win11, Inkscape 1.4.2.

 I suggest reporting this as a bug.

Inkscape remembers the dialog location on program close/open, but not when in session close/open the dialog(s).