Upgrade StorageZotero 6.* fails to update as non-admin on macOS 13.5, as Zotero 5.* did at least back to macOS 11.1
In keeping with the principle of least privilege [1] and Apple's recommendations [2], I always use a non-administrative account to operate my computers, authenticating as my administrative account only as necessary.
A long-lived bug in Zotero for Mac [3] causes its self-updater to fail in such cases (unlike, to name a few, Brave, Chalk, Firefox, GIMP, GPG, Hex Fiend, all 5 Mac TeX apps, QGIS, RStudio, Transmission, or Wireshark). Moreover, the mode of failure leaves the user with two instances of Zotero running concurrently, one of which is non-responsive, and necessitates force-quitting the one, quitting the other, switching to an admin account, installing the update, logging out of the admin account and back into the non-admin account, and, to prevent repeated failures to install the already-installed update on every subsequent launch of Zotero, deleting the cached update from a sub-directory of the (invisible by default) user Library folder.
To fix this, do not attempt to install the update until after any necessary administrative escalation is completed, and do not re-launch Zotero until after the update is installed. Additionally, ensure the update is downloaded to a directory the user-chosen administrator can read and write (so, not one within the non-admin user's Library folder), but not one any other user can write. (The administrator's Library/Caches/Zotero folder would fit the bill here.)
[1] https://www.beyondtrust.com/blog/entry/what-is-least-privilege
[2] https://support.apple.com/en-ca/guide/mac-help/flvlt003/mac
[3] https://forums.zotero.org/discussion/86790/v5-0-93-fails-to-properly-upgrade-on-macos-11-1-under-non-admin-user
A long-lived bug in Zotero for Mac [3] causes its self-updater to fail in such cases (unlike, to name a few, Brave, Chalk, Firefox, GIMP, GPG, Hex Fiend, all 5 Mac TeX apps, QGIS, RStudio, Transmission, or Wireshark). Moreover, the mode of failure leaves the user with two instances of Zotero running concurrently, one of which is non-responsive, and necessitates force-quitting the one, quitting the other, switching to an admin account, installing the update, logging out of the admin account and back into the non-admin account, and, to prevent repeated failures to install the already-installed update on every subsequent launch of Zotero, deleting the cached update from a sub-directory of the (invisible by default) user Library folder.
To fix this, do not attempt to install the update until after any necessary administrative escalation is completed, and do not re-launch Zotero until after the update is installed. Additionally, ensure the update is downloaded to a directory the user-chosen administrator can read and write (so, not one within the non-admin user's Library folder), but not one any other user can write. (The administrator's Library/Caches/Zotero folder would fit the bill here.)
[1] https://www.beyondtrust.com/blog/entry/what-is-least-privilege
[2] https://support.apple.com/en-ca/guide/mac-help/flvlt003/mac
[3] https://forums.zotero.org/discussion/86790/v5-0-93-fails-to-properly-upgrade-on-macos-11-1-under-non-admin-user