To minimize this, ideally use PAM (as noted in other posts).
For passwords only use something like Syspass, set permissions and access to groups on a "need-to-know" only basis. The export and decryption procedure requires 2 passwords typicallly only held by the admin + owner(s).
For remote access, implement something like Apache Guacamole where all passwords/SSH credentials are on the server and user accounts are audited and can be suspended at once.
Keep in mind passwords can still be copied manually given enough time.
Indeed, this may be a useful feature for some password managers. I use sysPass among others but only accessible via VPN, so it wouldn't be practical to "share this password" from it.
Yes, work related but with external providers and customers - so not something worth their time setting up the "right" way (for example OpenPGP encryption), sometimes even if they wanted to they can't because of their own policies ("can't install anything here" etc.).
I thought I'd share my positive experience with sysPass. I've filed several bug reports and had great response, and now after ~1y of using it I can fully appreciate easy updates, good community support and most importantly control of the data it stores.
If you're using sysPass consider making a donation!
I use SysPass for shared password management (has 2FA, mobile support, fine-grain access control) and Redmine as project tracker and wiki (per-project wiki, fine-grain access controls, IT/support/project tracker).
Both are free open source software.