Some question are you manually mounting the drive every time instead of auto-mounting it?
Because that makes a big difference. If you leave “User Session Defaults” enabled (or configure it to auto-mount at startup), the system mounts the drive automatically when you log in or when the PC starts. That means GNOME treats it as already trusted and you normally won’t get a password prompt. I’ve never had to change anything from /mnt to /run because my drives are auto-mounted, so I honestly didn’t think about that angle at first.
If you manually mount a drive especially if it is mounted under /mnt as a system/root mount GNOME may ask for a password. That’s not really about the folder name itself; it’s about
how the drive is mounted and which permissions/policies apply.
Difference between /mnt and /run/media in practice:
/mnt is traditionally used for manual or administrator-style mounts. Many times these are mounted as root (via sudo, scripts, or manual commands). When GNOME sees a root-owned mount that wasn’t created as a user session mount, it may require authentication (polkit) for actions like mounting, unmounting, or accessing certain operations.
/run/media/username/... is the default location used by udisks (the GNOME mounting system). These mounts are tied to your logged-in user session. Ownership and permissions are automatically set for the active user, so normally no password is needed. The system assumes it belongs to you because it was mounted through the desktop session.
That’s why what I gave you with chown should still be valid for ext4 ext4 supports real Linux ownership but if the mount itself is handled through GNOME with authorization enabled, ownership alone won’t remove the password prompt. The mount policy still applies.
If you want to configure auto-mount using GNOME Disks (like in the screenshot):
Open GNOME Disks → select the partition → click the gear icon → “Edit Mount Options”.
Then:
- Disable “User Session Defaults”.
- Enable “Mount at system startup” (or automatic mounting).
- Make sure “Require additional authorization to mount” is OFF.
- Set a stable mount point if you want (for example /mnt/mydrive or another folder).
- Give it a clear name if needed so it’s easy to recognize.
After that, the system should mount it automatically without asking for a password, similar to how mine behaves.
So the main difference here is not really /mnt vs /run, but manual mount vs auto/user-session mount and whether polkit authorization is required.