Just in case this helps anyone. I haven't heard back from Kaspersky support regarding my ticket, but I decided to try uninstalling the most recent Windows Update that was installed. After that uninstall completed (and the system rebooted), I can copy/paste passwords from the Kaspersky Password Manager again.
The update I uninstalled is: 2023-09 Cumulative Update Preview for Windows 11 Version 22H2 for x64-based Systems (KB5030310)
(I sent an update to my ticket with support as well, asking them to verify if they can replicate this)